MATH 1

Module 8 Powerful “O” What this module is all about This module will focus on a special kind of algebraic expressions calledpolynomials. You will perform operations on polynomials and solve problems involvingpolynomials. This module contains the following lessons: Lesson 1 Polynomials Lesson 2 Addition and Subtraction of Polynomials Lesson 3 Multiplication of Polynomials Lesson 4 Division of Polynomials Lesson 5 Application of the Operations on Polynomials What you are expected to learn After going through this module, you are expected to:• define polynomials;• classify algebraic expressions as polynomials and non-polynomials;• perform operations on polynomials  addition and subtraction  multiplication: polynomial by a monomial  multiplication of polynomial by another polynomial  division of polynomials by monomial and by another polynomial; and• solve some problem involving polynomials. 1

How to learn from this moduleThis is your guide for the proper use of the module: 1. Read the items in the module carefully. 2. Follow the directions as you read the materials. 3. Answer all the questions that you encounter. As you go through the module, you will find help to answer these questions. Sometimes, the answers are found at the end of the module for immediate feedback. 4. To be successful in undertaking this module, you must be patient and industrious in doing the suggested tasks. 5. Take your time to study and learn. Happy learning! The following flowchart serves as your quick guide in using this module. Start Take the Pretest Check your paper and count your correct answers. Is your score Yes Scan the items you 80% or above? missed. No Proceed to the nextStudy this module module/STOP.Take the Posttest 2

What to do before (Pretest)Direction: Choose the letter of the correct answer.1. Which of the following algebraic expressions is a polynomial?a. 3 − 2 c. 3a-2+1 xb. y + 1 d. x2+32. Which of the following is not a monomial?a. 3(2x+y) c. 2x2yb. (xy)3 d. 3x+y3. What is the degree of the monomial 3xy2z3 ?a. 1 c. 5b. 3 d. 64. What must you add to 3x2-2x+1 to get -x2+x-2?a. –4x2+3x-3 c. x2-x-1b. –4x2+3x-3 d. 2x2-x+15. What must be subtracted from a3-2a2+3a+4 to get 2a3+4a2+6a+8a. –a3-6a2- 3a - 4 c. a3-2a2+3a-4b. a3+2a2+3a-14 d. a3-2a2+3a+46. Find the product of this expression 2x(2x + 3y – 4).a. 4x2 + 6xy – 8x c. 2x2 – 6xy – 8b. 4x + 6y – 8 d. 4x2 – 6xy – 8x7. Divide 32ay2 – 16a2y3 + 8ay by 4ay.a. 8y – 4ay2 + 2 c. 8y3 + 4ay + 2b. 7y – 3ay + 2y d. 8y2 – 4a2y2 + 2a2y28. Find the area of the rectangle if its length is 3x + 2 and the width is x+2. 3

a. 4x2 + 6x + 4 c. 3x2 – 8x + 4b. 3x2 + 8x + 4 d. 4x2 + 8x + 49. The area of a rectangular garden is 4x2+2x. If its width is 2x, what is its length?a. 2x+2 c. x+1b. 2x+1 d. x+210. Which of the following statements is true about algebraic expressions and polynomials?a. Some polynomials are algebraic expressions.b. All polynomials are algebraic expressions.c. All algebraic expressions are polynomials.d. Algebraic expressions are special types of polynomials. Answer Key on page 33 What you will do Read the following lessons carefully and do the suggested activities patiently.Lesson 1 PolynomialsExplorationConsider the following examples and non-examples of polynomials.All of these are polynomials. 3x2y 6 2x3 − 6 4a2+3a+1 4

All of these are NOT polynomials. x 1 x 1 3 2x-2 a + 6 3 x y3 What is a polynomial?Did you know?Remember A polynomial is a special kind of algebraic expression where each termis a constant, a variable, or a product of constants and variables raised to wholenumber exponents. An algebraic expression is not a polynomial when• the variable is in the denominator, such as 3 , x xy• the exponent of the variable is not a whole number, such as 2x-2, 1 x 1 3 3• the variable is under a radical sign, such as a + 6 Can you identify which of the following algebraic expressions are polynomials byplacing a check mark before the number? _____1. 4x-y _____2. 3a b _____3. 4a2b2 _____4. 3x + 1 _____5. 4x2+2x+1 Did you check 1, 3 and 5? You are right! 5

A better way to describe these polynomials is by identifying them according to thenumber of terms they contain. • 4x – y is a polynomial of two terms. What is the special name for this polynomial?_________ • 4a2b2 is a polynomial of one term. What is the special name for this polynomial?_________ • 4x2 + 2x + 1n is a polynomial of three terms. What is the special name for this polynomial?__________ If you answered binomial, monomial and trinomial respectively, then you’re correct! Remember There are special names for polynomials according to the number of terms. A monomial has one term. A binomial has two terms. A trinomial has three terms. A multinomial has four or more terms. The degree of a monomial or each term of a polynomial is the exponent of itsvariables or the sum of the exponents of its variables. The degree of the polynomial is thehighest degree of its term or the highest degree of a monomial. Study the following examples in determining the degree of polynomials. • The degree of –8 is zero, since –8 = -8x0 which is a constant term. • The degree of y2 is 2 in variable y. • The degree of 3xy is 2 in variables x and y. • The degree of 4x4-3x3+2x2+1 is 4 in variable x. • The degree of x3y-x2y+xy2 is 4 in variables x and y. Can you tell the degree of each polynomial? • 6x, the degree is ______ in variable_____ • 7x2y, the degree is_____ in variables____ • 3a+4a2b2, the degree is _____ in variables____ • 3x2+4x3-5x5, the degree is____ in variable____ Are your answers correct? Compare them with my answers. 6

My answers: The degree of 6x is 1 in variable x. The degree of 7x2y is 3 in variables x and y. The degree of 3a+4a2b2 is 4 in variables a and b. The degree of 3x2+4x3-5x5 is 5 in variable x. Study the example below. The polynomial 3x2 + 4x3 - 5x5 is a polynomial in one variable x. Arrange the terms indescending powers of the variable x._____ _____ ____1st 2nd 3rd Did you write –5x5 + 4x3+3x2? See, you can do it! When you arrange polynomials in one variable in descending powers of its variables,you can easily determine the degree of the polynomials.Remember The degree of a polynomial in one variable is the value of thelargest exponent of the variable that appears in any term. Could you arrange the polynomials in descending powers of its variable? ____Try. You can do it!Polynomials Descending Order Degree• -3x2+6x3 _____________________ ________• b -3b2• m – 3m2+ 5m3 ______________ _____• -8 + 6x8 ______________ _____• 14a6 +14+ 14a ______________ _____ ______________ _____Compare your answers with mine. Check if you got the correct answers. 7

My answers: Descending Order Degree 6x3-3x2 3 Polynomials 2 -3x2+6x3 -3b2 +b 3 b -3b2 5m3– 3m2 +m 8 m – 3m2+ 5m3 6x8-8 6 -8 + 6x8 14a6 +14a+14 14+14a+14a6Self-check 1Directions: Answer as indicated.1. The expression 3 + 3y2 is not a polynomial. (True, False) x2. What is the degree of –10?a. –1 b. 0 c. 1 d. 103. Arrange the following terms in descending order by numbering 1, 2, 3, 4…2m -4 3m2 m_____ _____ _____ _____4. Decide whether the polynomial in number 3 is a monomial, binomial, trinomial or multinomial. __________5. What is the degree of the polynomial in number three? ___________ Compare your answers with those in the Answer key. If you got all the answerscorrect, CONGRATULATIONS! You did well in the lesson and can proceed to the nextlesson. If not, you need to go over the same lesson and take note of the mistakescommitted. It pays to have a second look. Answer Key on page 33 8

Lesson 2 Addition and Subtraction of PolynomialsExploration To add and subtract polynomials, you must recall the difference between like andunlike terms.Study the examples of like terms and unlike terms. Like terms Unlike terms 2, -5, 7.5 3, y x, 8x, − 2 x 3x2, 3 x3 3 4 3xy2, -4xy2 3xy2, 3x2yDetermine which of the following are pairs of like terms by placing a check before thenumber. _____1. 25b, 25b2 _____4. a4b2c, -11a4b2c _____2. –3x3y2, 2x3y2 _____5. –0.4, 2 3 _____3. –0.5c, 1 c _____6. 3x5y, 3xy5 3Did you check 2, 3, 4 and 5? You’re correct! Remember Like terms or similar terms are monomials that contain the same literal coefficients, that is, the terms have exactly the same variables and exponents.A polynomial is in the simplest form when all like terms are combined. Study and learn from the illustration below. 9

Simplify1. 11t + 7t = (11+7)t =18t2. 2xy-5xy+4xy= (2-5+4)xy = xy3. 2x2-5x+6x-x2+11= (2x2-x2) + (-5x+6x) +11 Group like terms = (2-1)x2 +(-5+6)x + 11 = x2 + x + 11What property is applied in simplifying polynomials?_________Did you answer distributive property? You’re right!Remember • Distributive Property: a(b+c) = ab + bc • In simplifying polynomials, combine like terms by adding or subtracting their numerical coefficients and multiplying the result by the common literal coefficient. Your skills in combining like terms can be used in adding and subtractingpolynomials.Consider these examples:1. Add x2-2x + 3 and 4x2+x-2 Grouping like (x2-2x + 3) + (4x2+x-2) = (x2+4x2) + (-2x + x) + (3-2) terms = (1+4)x2 + (-2+1) x +(3-2) Distributive = 5x2 – x + 1 property2. Add: ( 3 – 2x + x2) + (6x2 + 5x –4) A vertical arrangement may be used to add polynomials. Like terms are written in thesame column. x2 – 2x + 3 10

6x2 + 5x - 4 7x2 + 3x –1 Since subtraction is defined as addition of the opposite or additive inverse,subtraction is very similar to addition of polynomials.3. Subtract:: (4x2-4x) – (7x2 – 3x)(4x2-4x) – (7x2 – 3x) = ( 4x2 - 4x) + ( -7x2 + 3x) Add the additive inverse = (4x2 – 7x2) + (-4x + 3x) Grouping like terms = (4-7)x2 + (-4+3)x Distributive property = -3x2 – x4. Subtract 4a3 –2a2 + 6a – 5 from 6a3 –5a2 + 11a + 8 What is the minuend? ______________________ What is the subtrahend? ____________________ Check your answers with mine.My answer:The minuend is 6a3 –5a2 + 11a + 8The subtrahend is 4a3 –2a2 + 6a – 5 The mere identification of the minuend and the subtrahend facilitate accuracy insubtracting polynomials.Subtracting polynomials vertically, 6a3 –5a2 + 11a + 8 6a3 –5a2 + 11a + 8- 4a3 –2a2 + 6a – 5 + -4a3 +2a2 –6a + 5 2a3 – 3a2 + 5a + 13 11

Self-check 2Do the indicated operation:1. Find the sum : 8x4 – 5x3 + 15x2 – 7 -3x4 + 6x3 – 8x2 – 2 14x4 + x3 – x2 + 32. Find the difference: (25m3 + 6m2 – 7m) – (9m3 + 12m2 + 4m)3. Add: ( -4a + ab + 3b) + (2ab – 8b – 6a) + ( 3 + a – ab)4. Subtract 30x2y – 15xy + 6y from (25x2y + 9xy – 7y). Compare your answers with those in the Answer key. If you got all the answerscorrect, CONGRATULATIONS! You did well in the lesson and can proceed to the nextlesson. If not, you need to go over the same lesson and take note of the mistakescommitted. It pays to have a second look. Answer Key on page 33Lesson 3 Multiplication of PolynomialsExploration In multiplying polynomials, the distributive property and the laws of exponents areused extensively. Observe how these two concepts are used in the following illustrations.1. 2x(x + y – 2) = 2x(x) + 2x(y) + 2x(-2) Distributive property 12

= 2x2 + 2xy – 4x am an = am+n that is, 2x ⋅ x = 2x 22. 3a2(a2 + 2ab + 3b2 – 3) = 3a2(a2) + 3a2(2ab) + 3a2(3b2) + 3a2(-3) = 3a4 + 6a3b + 9a2b2 – 9a2Try this one.Find the product of 3n(2m2 - 3mn + 4)• How many terms are there in the product?______• The first term is the product of 3n(2m2) = _______• The second term is the product of 3n(-3mn) = _____• The third term is the product of 3n(4) = ______If you answered…• There are three terms in the product.• The product is 6m2n – 9mn2 + 12nThen you are correct!Remember The distributive property is used to multiply a polynomial bya monomial. The monomial is distributed over each of the terms ofthe polynomial. The laws of exponents, am an = am+n is used to find theproduct of each monomial. To multiply binomial by another binomial, use the distributive property twice. Observehow it is done in the illustration below.1. Multiply (4a+b) ( x – 2y)(4a+b) ( x – 2y) = 4a(x –2y) + b( x –2y) Distributive property = 4a(x) + 4a(-2y) + b(x) + b(-2y) Distributive 13

property = 4ax – 8ay + bx – 2by To help you remember which terms to multiply in multiplying two binomials a simplememory aid called FOIL is used . FOIL stands for F(First), O(Outer), I(Inner) and L(Last).See below how this FOIL method is used. 2. Multiply ( 2x –3y) (x-2y) F: ( 2x –3y) (x-2y) Multiply first terms: 2x2 O: ( 2x –3y) (x-2y) Multiply outer terms : -4xy I: ( 2x –3y) (x-2y) Multiply inner terms : -3xy L: ( 2x –3y) (x-2y) Multiply last terms : 6y2 The result is 2x2-4xy –3xy + 6y2. Combining like terms, the answer is2x2 – 7xy + 6y2.Try this one.3. Multiply (3x-4) (4x + 5) using the FOIL method. F: ___________________ O: ___________________ I: ____________________ L: ____________________ The product is ____________Compare your answer with my answer.My answers:F: 12x2 O: 15x I: -16x L: -20The product is 12x2-x –20 by combining like terms.Remember The FOIL method is a simple memory aid for multiplyingbinomials using the distributive property. 14

The distributive property can also be used to multiply polynomials of any number ofterms. Observe how this is done below.3. Multiply ( x - 4) ( x2 – 5x + 4)(x - 4) ( x2 – 5x + 4) = x(x2 – 5x +4) – 4(x2 – 5x +4) Distributive = x3 –5x2+ 4x – 4x2+ 20x – 16 property = x3 –9x2 + 24x –16 Distributive property Combining like termsTry this one.4. Multiply (2x +5) (x2 – x – 1) (2x +5) (x2 – x – 1)= _________________ =__________________ =___________________Compare your answer with my answer.My answer: (2x +5) (x2 – x – 1) = 2x(x2 – x – 1) + 5(x2 – x – 1) = 2x3 –2x2 –2x + 5x2 –5x –5 = 2x3 + 3x2 –7x – 5 Another way to do it is by the vertical method. Study and learn from the followingillustration. 15

5. Multiply (2x + 7) (3x – 2) by vertical method. 2x + 7 Multiply 2x +7 by -2 3x – 2 Multiply 2x + 7 by 3x -4x –14 Add6x2+ 21x6x2 + 17x –14Try this. 6. Multiply (2x + 5) (x-2) by vertical method. 2x + 5 x-2Compare your answer with mine.My answers: 2x + 5 x-2 -4x – 10 2x2 + 5x 2x2 + x –10Remember In using the vertical method, make sure to include thesign of the term in multiplying that term to each of the terms inthe first factor. 16

Self-check 3Multiply. You may use any style of multiplying the polynomials. 1. 3abc(-4abc + 2a2b2c2 – 3a3b + c) 2. (3x+2) (5x – 4) 3. (3x –2) (x +4) 4. (5a2 – 6a + 3) (2a + 5) 5. ( x2 – 4x + 2) ( x – 4) Compare your answers with those in the Answer key. If you got all the answerscorrect, CONGRATULATIONS! You did well in the lesson and can proceed to the nextlesson. If not, you need to go over the same lesson and take note of the mistakescommitted. It pays to have a second look. Answer Key on page 33Lesson 4 Division of Polynomials Exploration The laws of exponents are used when one monomial is divided by another. Below isan illustration on how these laws are used. 17

Divide the following. a m ÷ a n = a m−n .......if ...m > n1. 25 ÷ 23 = 25−3 = 222. x4 ÷ x7 = 1 = 1 am ÷ an = 1 .......if ...m < n x 7−4 x3 a n−m3. y 6 ÷ y 6 = y 0 = 1 a m ÷ a n = a0 = 1.......if ...m = nAnother way to answer 2 is shown below. x 4 ÷ x 7 = x 4−7 = x −3Thus, x-3 = 1 . x3 Remember • Definition of a Negative Exponent For a ≠ 0,..a −n = 1 an • Laws of Exponent a m ÷ a n = a m−n .......if ...m > n am ÷ an = 1 .......if ...m < n a n−m a m ÷ a n = a0 = 1.......if ...m = n You can now use these laws and the definition together with the operations on realnumbers in dividing monomials. Study and learn from the following illustration. Divide the following monomials. 18

1. 48x3 ÷16x 2 = 48 x3−2 = 3x or 3(16)x ⋅ x ⋅ x = 3x 16 16 ⋅ x ⋅ x2. − 24xy3 ÷ 4x2 y5 = − 24 ⋅ 1 ⋅ 1 = −6 4 x 2−1 y 5−3 xy 23. 18ab 2c3d 4 = − 9 a1−2b 2−1c3−4d 4−3 = − 9 a −1bc −1d = − 9bd − 4a 2bc 4d 3 2 2 2acTry thisDivide the following.1. − 27xz 2 ÷ −9xz = ________________2. 81c4d 3 = ___________________ − 3c3d 4If you answered. . .1. − 27xz 2 ÷ −9xz = 3x1−1z 2−1 = 3x0 z = 3z2. 81c 4d 3 = −27c 4−3d 3−4 = −27cd −1 = − 27c − 3c3d 4 dThen you’re correct! Remember To divide a monomial by a monomial: 1. Divide the numerical coefficients. Express the quotient as a rational number in simplest form. 2. Apply the laws of exponent and make all exponents positive. To divide a polynomial by a monomial, you may use the distributive property. See theillustration to understand this task. Divide. 19

x − 3x2 + 4x − 6 = x3 − 3x2 + 4x − 6 x x x xx1. Distributive Property = x2 − 3x + 4 − 6 Dividing each monomial x2. 28a3b5 − 21a 4b3 + 49a3b6 = 28a3b5 − 21a 4b3 + 49a3b6 Distributive Property 7a 2b 3 7a2b3 7a2b3 7a2b3 = 4ab2 − 3a 2 + 7ab3Try this.Divide −12x 2 y 2 − 20x 2 y + 4x . − 4xCompare your answer with mine.My answer is 3xy2+5xy – 1. Remember To divide a polynomial by a monomial 1. Divide each term of the polynomial by the monomial. 2. Add the resulting quotient. Be sure to indicate the sign before each term using the rule of sign. The process of dividing a polynomial by a polynomial is similar to the long divisionprocess in arithmetic. Look at the illustration below for this comparison. • Divide 107 by 3. Divisor 35 Quotient 3 107 Dividend Divide: 10 ÷ 3 9 Multiply: 3X3 17 15 Subtract. Bring down the next digit. Repeat the steps. Remainder 2 Stop when the remainder is 0 or it is less than the divisor. 20

Thus, 107 = 35 + 2 . 33Generally, you write dividend = quotient + remainder divisor divisor• Divide: (8a 2 + 10a − 7) ÷ (2a −1) 4a Divide 8a2 by 2a2a-1 8a 2 + 10a − 7 Multiply 2a –1 by 4a 8a2- 4a Subtract and bring down the next term. 14a – 7 4a + 7 Divide 14a by 2a2a-1 8a 2 + 10a − 7 Multiply 2a – 1 by 7 8a2 –4a 14a – 7 14a – 7 0 Stop when the remainder is 0.Thus, the quotient is 4a + 7. Remember 1. Before dividing, arrange the terms in descending power of the variables. 2. Insert 0 for the missing terms of the dividend or the divisor.Try this.Divide (x3-13x-12) by (x-4).Did you answer this way? x2 + 4x + 3 Insert 0x2x-4 x3 + 0x 2 −13x −12 x3- 4x 2 4x2- 13x 4x2 – 16x 3x- 12 21

3x – 12 0 The remainder is 0. Thus, x3 −13x −12 = x2 + 4x + 3 x−4 Then you are right! Self-check 4Divide.1. 2a3b2 − 10a 4b3 5a 2b 22. 9m4n3 + 15m3n4 − 21m2n2 3m 2 n3. x + 5 x2 + 7x + 34. x − 6 3x2 −14x − 24 Compare your answers with those in the Answer key. If you got all the answerscorrect, CONGRATULATIONS! You did well in the lesson and can proceed to the nextlesson. If not, you need to go over the same lesson and take note of the mistakescommitted. It pays to have a second look. Answer Key on page 33 22

Lesson 5 Application of the Operations on PolynomialsExploration The skills developed in performing operations on polynomials would be moremeaningful if you can solve problems. Furthermore, solving word problems will develop yourreasoning and thinking power.Do the activity below.Can you identify the figures below?12 cm 3 cm5cm 2cm 3cm(1) (2) 4 cm (3)• Figure 1 is _________• Figure 2 is _________• Figure 3 is _________Did you answer rectangle, square, and triangle? You’re right!Can you find the perimeter of each figure?• The perimeter of figure 1 is _____ cm• The perimeter of figure 2 is _____ cm• The perimeter of figure 3 is _____cm 23

• How do you compute for the perimeter of each of these figures? ______________________________________________________Compare your answers with mine.My answers: • The perimeter of figure 1 is 5+5+12+12= 2(5)+2(12)= 34 cm • The perimeter of figure 2 is 3+3+3+3= 4(3)= 12 cm • The perimeter of figure 3 is 2+3+4= 9cm • How do you compute for the perimeter of each of these figures? I find the perimeter of each figure by adding the measure of each side. Remember The perimeter is the sum of the measure of the sides of any polygon. • A rectangle is a four-sided polygon with two pairs of opposite sides are congruent and having four right angles. • A square is a polygon with four sides congruent and having four right angles. • A triangle is a three-sided polygon. It can be a scalene triangle (no sides congruent), an isosceles triangle (2 sides congruent), or an equilateral triangle (3 sides congruent). Your knowledge in combining similar terms can be used to find formulas for theperimeter of these polygons. L W s ab (1) (2) cFind the perimeter of each figure above. (3)24

• The perimeter of (1) is ______ units • The perimeter of (2) is ______ units • The perimeter of (3) is ______ unitsIf you answered. . . • The perimeter of (1) is L+L+W+W= 2L+ 2W units • The perimeter of (2) is s+s+s+s = 4s units • The perimeter of (3) is a+b+c units.Then you’re correct!Notice that these are the formulas for the perimeter of these polygons. Remember The perimeter of a rectangle = 2L + 2W, where L is the length and W is the width. The perimeter of a square = 4s, where s is the length of sides. The perimeter of a triangle= a+b+c, where a, b and c are the sides. You can also apply your skills in addition of polynomials in finding the perimeter ofthese polygons if the sides are polynomials.2x + y x2-xx+y 3x+1 (1) (2) 4x-3P= _________ P=_________ (3) P=_________ 25

Compare your answers with mine.My answers: (1) P = 2(x+y) + 2(2x+y) = 2x+2y + 4x + 2y = 6x + 4y (2) P = 4(x2=x) = 4x2- 4x (3) P= (3x+1) +(3x+1) +(4x-3) = 10x -2 You are now ready to solve problems about perimeter using your skills in performingoperations on polynomials. Read and analyze this problem below. Problem: The length of a rectangle is 3 times greater than its width. Write an expression to represent its perimeter. If you represent x = width, how will you represent the length? _____ Did you answer 3x? You’re right. Since the length is 3 times greater than the width. What formula should you use to answer this problem? ___________ Yes, the perimeter of the rectangle P= 2L +2w, where L is the length and W is thewidth. How will you solve this problem?___________ Compare your solution with mine. My solution: P= 2L + 2W = 2(3x) + 2(x) = 6x + 2x = 8x units Do the activity that follows. 26

Find the area of the following figures. 12 cm5 cm 3 cm 3 cm (1) (2) 6 cm • The area of figure 1 is _________ sq cm (3) • The area of figure 2 is _________ sq cm • The area of figure 3 is _________ sq cmCompare your answer with my answer.My answers: • The area of figure 1 is (5cm) (12cm)= 60 sq cm • The area of figure 2 is (3cm)(3cm)=(3cm)2= 9 sq cm • The area of figure 3 is sq cm 1 (6cm)(3cm) = 9 sq cm 2How do you find the area of these figures? Did you use these formulas? Remember • Area of a rectangle is length times the width: A=LW • Area of a square is the square of its side: A= s2 • Area of a triangle is one-half times its base and height: A = 1 bh 2 27

You can also apply your skills in multiplying of polynomials in finding the areas ofthese polygons if the sides are polynomials. Find the area of the following figures. 6x3x 6xy 3y (1) (2) 8yA =________ sq units A = ______ sq units (3) A = _____sq unitsCompare your answers with mine.My answers:(1) A= 3x(6x)= 18x2 sq units(2) A = (6xy)2= 36x2y2 sq units(3) A = 1 (3y)(8y) = 12y2 sq units 2Read and analyze the problem and answer the questions that follow.Problem: One side of a square is 2x+3, find the area of the square. • What is 2x + 3 in the problem? _______ • What is asked in the problem? _______ • What formula can you use? ______ • What is your answer? ________ Compare your answer with my answers. 28

My answers: • 2x + 3 is the side of the square • the area of the square • A = s2 • A= (2x+3)2= 4x2+12x+9 sq units Self-check 51. A triangle has 3 sides. The first side is x + 5m, second side is x + 7m and the third side is 2x + 3m. Find the sum of the lengths of the sides of the triangle.2. One side of a square is 2x + 2. Find the area of the square. Follow this formula: A = (s)2.3. A garden lot has an area of 8a2 + 10a – 63. If the width is 2a + 7, find the length. Answer Key on page 33 29

Let’s summarize• A polynomial is a special kind of algebraic expression where each term is a constant, a variable, or a product of constants and variables raised to whole number exponents.• There are special names for polynomials according to the number of terms. A monomial has one term. A binomial has two terms. A trinomial has three terms. A multinomial has four or more terms.• The degree of a polynomial in one variable is the value of the largest exponent of the variable that appears in any term.• Like terms or similar terms are monomials that contain the same literal coefficients, that is, the terms have exactly the same variables and exponents.• In simplifying polynomials, combine like terms by adding or subtracting their numerical coefficients and multiplying the result by the common literal coefficient.• The distributive property is used to multiply a polynomial by a monomial. The monomial is distributed over each of the terms of the polynomial.• The laws of exponents, am an = am+n is used to find the product of each monomial.• To divide a monomial by a monomial, i). Divide the numerical coefficients. Express the quotient as a rationalnumber in simplest form. ii) Apply the laws of exponent and make all exponents positive.• To divide a polynomial by a monomial i). Divide each term of the polynomial by the monomial. ii). Add the resulting quotient. Be sure to indicate the sign before each term using the rule of sign. 30

What to do after (Posttest)Choose the letter of the correct answer.1. In an algebraic expression, the degree of a monomial is the sum of the _____ ofits variables.a. base c. termsb. exponents d. constant2. Of the given expressions, which is a polynomial?a. 6x + 3 c. 10b. 3m2 – 3 x2 d. m-2m3. Look for the similar terms in the given expressions.a. 2b, 2a, -3ab c. 6ac, 4a2c, 7ac2b. 8x, -3x, 4x d. 4x2, 3x, 2ax4. What must you add to x2+ 3x+ 4 to get 2x2 – 5x + 6?a. x2-8x+2 c. 2x2 +2x + 2b. –x2 + 8x + 2 d. -2x2 –2x -105. What must be subtracted from 2a3+ 4a2+6a + 8 to get a3-2a2+3a+4?a. –a3 + 6a2+3a + 4 c. a3-2a2+3a-4b. a3+2a2+3a-14 d. a3-2a2+3a+46. Multiply the polynomials: 2x2(2x + 3y – 4)a. 4x2 + 6xy – 8x c. 4x3 + 6x2y – 8x2b. 4x3 + 6xy – 8 d. 4x3 – 6x2y + 8x27. Divide 4y2 + 29y + 7 by y + 7.a. y – 7 c. y + 7b. 2y – 7 d. 4y + 18. Find the perimeter of a square whose side is 3x-1.a. 12x – 4 c. 9x2-1b. 12x – 1 d. 9x2 –6x + 1 31

9. Find the area of the rectangle if its width is x + 2 and its length is 2x + 6.a. 2x2 + 10x – 12 c. 2x2 + 12x + 10b. 2x2 + 10x + 12 d. x2 + 10x + 1210. Which of the following statements is NOT true about algebraic expressions and polynomials? a. Some algebraic expressions are polynomials. b. All polynomials are algebraic expressions. c. All algebraic expressions are polynomials. d. Polynomials are special type of algebraic expressions. Answer Key on page 33 32

Answer KeyPretest page 31. c 6. a2. d 7. a3. d 8. b4. a 9. b5. a 10. bLesson 1 Self-Check 1 page 8 Lesson 5 Self-Check 1 page 29 1. 4x + 15m units1. True 2. 4x2+ 8x + 4 sq units 3. 4a – 9 unit2. 0 m3. 2m -4 3m2 3 14 24. multinomial5. 2Lesson 2 Self-Check 2 page 12 Posttest page 31 1. 19x4 + 2x3 + 6x2 – 6 1. b 6. 6 2. 16m3 – 6m2 – 11m 2. a 7. d 3. –91 + 2ab – 5b + 3 3. b 8. a 4. –5x2y + 24xy – 13y 4. a 9. b 5. a 10. cLesson 3 Self-Check 3 page 171. -12a2b2c2 + 6a3b3c3 – 9a4b2c + 3abc22. 15x2 – 2x - 83. 3x2 + 10x - 224. 10a3+13a2 – 24a + 155. x3 – 8x2+ 18x - 8Lesson 4 Self-Check 4 page 22 1. 2 a − 2a 2b 5 2. 3m2n2 + 5mn3 –7n 3. x + 2 − 7 x+5 4. 3x + 4 33

Module 9 The R and S in Math What this module is all about This module is about mathematical phrases, sentences, first-degree equations andinequalities. In the English language, letters and punctuations are used to create words andphrases. In Algebra, letters along with numbers and symbols of operations are used tocreate expressions. Expressions together with the relation symbols are used to createequations and inequalities. Such equations and inequalities are used to model and solvereal-life problems. You will learn more about these concepts as you study the four lessonsin this module. Lesson 1 Mathematical Phrase, Mathematical Sentences, Equations and Inequalities Lesson 2 Translating Verbal Statements to Equations or Inequalities and vice-versa Lesson 3 Differentiating First-degree Equations from First-degree Inequalities in One Variable Lesson 4 Applications of Equations and Inequalities What you are expected to learn After working on this module, the student is expected to: • distinguish between  mathematical phrases and mathematical sentences;  equations and inequalities; • translate verbal sentences to equations or inequalities and vice-versa; • define first degree equations and first-degree inequalities in one variable; and • apply equations and inequalities in some real-life situations. 1

How to learn from this moduleThis is your guide for the proper use of the module: 1. Read the items in the module carefully. 2. Follow the directions as you read the materials. 3. Answer all the questions that you encounter. As you go through the module, you will find help to answer these questions. Sometimes, the answers are found at the end of the module for immediate feedback. 4. To be successful in undertaking this module, you must be patient and industrious in doing the suggested tasks. 5. Take your time to study and learn. Happy learning! The following flowchart serves as your quick guide in using this module. Start Take the Pretest Check your paper and count your correct answers. Is your score Yes Scan the items you 80% or above? missed. No Proceed to the nextStudy this module module/STOP.Take the Posttest2

What to do before (Pretest)Multiple Choice. Choose the letter of the correct answer.1. Which of the following is a mathematical phrase?a. x2 + 1 < 10 – 11 c. x – 1 > 10b. 3m2 + 2n – 16 = 0 d. x2 + 4x + 12. Which of the following is NOT an equation?a. P = 21 = 2w c. x2 – 3x > x –2b. x2 – 1 = 0 d. x2 + 7x = -103. Which of the following is a mathematical sentence? i. 2(x – 5) > 1 ii. a + 4 = 7 – 2aa. i b. ii c. i and ii d. none of the above4. Which among the following is NOT an open sentence?a. 45n + 7 = 3 c. 18 > 2b. c – 4 = 0 d. –3 ≤ -½5. What is the equivalent mathematical sentence for “The sum of twice a number and five is equal to thirteen”?a. 2x + 5 = 13 c. 2(5) + x = 13b. 2(x + 5) = 13 d. 2x = 5 + 136. Translate this verbal sentence into an equation: “Two less than a number is twenty-one.”.a. 2y = 21 c. y = 21 - 2b. 2 – y = 21 d. y – 2 = 21 3

7. Which of the following is equivalent to 5x = 40?a. Five added to a number is forty.b. Five decreased by a number is forty.c. Five times a number is forty.d. The quotient of five and a number is forty.8. What is a verbal statement for the mathematical sentence 3a – 4 = 7?a. A number subtracted by four is seven.b. Twice a number subtracted by four is seven.c. Thrice a number subtracted from four is seven.d. Thrice a number subtracted by four is seven.9. The following are first-degree equations in one variable EXCEPTa. x2 +4 = 0 c. b + 9 = -3b. 3x – 4 = 10 d. 25a = 5010. Which of the following is a first-degree equation in one variable?a. 1/v2 – 3 = 4 c. x2 = 0b. ½ a = n d. 3n – 5 = r311. The following symbols are used in a first-degree inequality in one variable EXCEPTa. = b. < c. > d. ≤12. Which among the following is a first-degree inequality in one variable? i. d < 5 (6) ii. v3 > 0a. i b. ii c. i and ii d. none of the above13. What equation represents the distance d traveled by a car at the rate of 60 km/hr in 4 hours?a. d = 60 + 4 c. 4 = 60db. 60 = 4d d. d = 60 (4) 4

14. The amount of electricity consumed by Mark’s family is twice the amount consumed by Joy’s family. If Mark’s family consumed an amount x of electricity this month, what first-degree equation in one variable will be used to model this situation if the total amount consumed by them is P1050?a. x = 1050 c. 2x = 1050b. x + 2x = 1050 d. x = 1050 + 2x15. The perimeter p of a square is four times the length of its side s. If the perimeter is 120 cm., what first-degree equation in one variable will relate the given perimeter and the length of the side of the square?a. s = 120/4 c. s + 4s = 120b. 4 (120) = s d. 120 = 4s Answer Key on page 21 What you will doRead the following lessons carefully.Lesson 1 Mathematical Phrase, Mathematical Sentence, Equations and Inequalities In the previous module, you learned about algebraic expressions. Let us recall itsdefinition. An algebraic expression is a collection of constants and variables that are combined using one or more of the four fundamental operations namely, addition, subtraction, multiplication and division (except division by zero). 5

Exploration Let us now observe some algebraic expressions that are contained in themathematical phrases and mathematical sentences presented below. Mathematical Phrases Mathematical Sentences 20 – 12 20 – 12 = 8 2b 2b = 4 7=x+y x+y 5s = t 5s 3+4<0 -7c ≤ 1 3r + 4 e – 4d > ½ 2v + w ≥ 9a -7c e – 4d 2v + wWhat do you observe? How do you compare a mathematical phrase with a mathematicalsentence? _________________________________________________________________________________________________________You are correct! A mathematical phrase contains an algebraic expression that does notexpress a complete thought. On the other hand, a mathematical sentence containsalgebraic expressions together with a relation symbol =, <, ≤, > or ≥ and it expresses acomplete thought. We recall that these relation symbols are read as follows. = is equal to or equals < is less than ≤ is less than or equal to > is greater than ≥ is greater than or equal toNow, let us look at the given mathematical sentences. The first four mathematicalsentences namely, 20 – 12 = 8, 2b = 4, 7 = x + y and5s = t are called equations. Can you give your own examples of equations?________________________________________________________________If your answers contain algebraic expressions together with the sign =, then you are right.In your view, what is an equation? ____________________________________________________________________________________________________ 6

Good. An equation is a mathematical sentence that makes use of the symbol =. What doyou think does the symbol = imply? ____________________________Yes, the symbol = implies that the two sides of the equation are equal. This means thatwhatever is the value of the left side of the equation is also the value of the right side.This time, let us focus on the last four mathematical sentences namely, 3 + 4 < 0,-7c ≤ 1, e – 4d > ½ and 2v + w ≥ 9a. These mathematical sentences are calledinequalities. Can you give your own examples of inequalities?____________________________________________________________________________________________________________________________________If your examples contain algebraic expressions together with the relation symbols <, ≤, > or≥, then you are correct.How do you then define an inequality? _________________________________________________________________________________________________Very good. An inequality is a mathematical sentence that makes use of the relationsymbols <, ≤, > or ≥.What do the symbols < and > imply? ___________________________________________________________________________________________________Yes. Thesymbols < and > imply that the left side of the inequality is not equal to the right side of theinequality. This means further that the symbol < is used when the value of the left side ofthe inequality is less than the value of the right side, while the symbol > is used when thevalue of the left side of the inequality is greater than the value of the right side.The symbol ≤ means that the value of the left side of the inequality is either less than orequal to the value of the right side, while the symbol ≥ means that the value of the left sideof the inequality is either greater than or equal to the value of the right side. Again, let us go back to each of the given mathematical sentences and tell whether itis true or false. Mathematical sentence True or False? 1. 20 – 12 = 8 _____ 2. 2b = 4 _____ 3. 7 = x + y _____ 4. 5s = t _____ 5. 3 + 4 < 0 _____ 6. -7c ≤ 1 7. e – 4d > ½ _____ 8. 2v + w ≥ 9a _____ _____ 7

If your answer is true for the first mathematical sentence, false for the 5th mathematicalsentence, while may be true or false or neither true nor false, for the remainingmathematical sentences, then you are correct.Sentences 2, 3 4, 6, 7 and 8 may be true or false depending upon the value/s of thevariable/s. For example, in the equation 2b = 4if b = 2 then 2(2) = 4 and the equation is true,but if b = –1 then 2(-1) = 4 thus, the equation is false.Sentences 2, 3, 4, 6, 7 and 8 are examples of open sentences. An open sentence is anequation or inequality that becomes true or false when the variable is replaced by a value.Let us summarize what you learned in this lesson.Remember A mathematical phrase is an expression that does not express a complete thought. A mathematical sentence is an expression together with a relation symbol =, <, ≤, > or ≥. It expresses a complete thought. A mathematical sentence may be an equation or inequality. An equation is a mathematical sentence that makes use of the symbol =. An inequality is a mathematical sentence that makes use of the relation symbols <, ≤, > or ≥. An open sentence is an equation or inequality that becomes true or false when the variable is replaced by a value. 8

Self-check 1A. Tell whether each of the following is a mathematical phrase or a mathematical sentence.1. x + ½ + 2 6. 2x + 21 ≥ 42. 3(x – 2) = 2(x +5) 7. 4(c2 + d2)3. y – 5 ≤ 74. y + 1/y 8. 10 - x5. y2 + 5y 9. 11 + 5x < 3x -1 10. n – (n+2) = 13B. Consider the following mathematical sentences. Classify as true, false or open.1. One kilometer is equal to 1000 meters.2. It is the world’s largest archipelago.3. 5 + 9 =144. 9x + 4 = 205. 15 > 21C. Fill in the box with the relation symbol =, <, ≤, > or ≥.1. 3 + 5 9–12. 8 2(7 – 3)3. (39 ÷ 3) + 2 6 +2(5 – 1)4. x + 6 13, if x is replaced by 75. 2m 6 + 9, if m is 1 Answer Key on page 21 9

Lesson 2 Translating Verbal Statements to Equations or Inequalities and vice-versa A knowledge of mathematical symbols and their meanings will enable you totranslate verbal sentences into mathematical sentences and vice-versa. Let us study the chart below. Symbol Word/Phrase +added to, increased by, more than,the sum of, plus -Subtracted from/to, decreased by, ()diminished by, less than, the difference or sometimes not written anymore ÷, /, __As much as, of, as many as, product of =Divided by, the quotient of, ratio, over <Is equal to, equals, is the same as ≤is less than >is less than or equal to, at most ≥is greater thanis greater than or equal to, at least Now, let us use the phrases and their corresponding symbols given in the chart totranslate verbal sentences into equations or inequalities.Example 1. Translate each of the following into a mathematical sentences.1.1 verbal sentence: Three times a number is nine.translation: 3⋅ n =9mathematical sentence: 3⋅ n = 9 or 3n = 9 We note that the symbol for the operation multiplication may not be writtenanymore. 10

1.2 verbal sentence: The sum of a number and seven is twelve.translation: b+ 7 = 12mathematical sentence: b + 7 = 121.3 verbal sentence: The difference between a number and one is eight.translation: c- 1=8mathematical sentence: c–1=8 Let us see if you can do the same thing in the following sentences. Write thecorresponding symbols below.1.4 verbal sentence: A number added to six is greater than two.translation: ____ ___ ___ ___ __mathematical sentence: ________________________________________If your answer is x + 6 > 2, then you are correct.1.5 verbal sentence: Twice a number subtracted by nine is less than five.translation: ___ ___ __ ___ ___ ___mathematical sentence: _________________________________________ 11

If your answer is 2x – 9 < 5, then you got it right. What if you are to translate a mathematical equation into a verbal sentence? Let usconsider the following examples.Example 2. Translate each mathematical sentence into a verbal sentence.2.1 mathematical sentence: x y = 16verbal sentence: The product of x and y is sixteen. We note that the given mathematical sentence may also be translated as “Theproduct of a number and another number is 16.” or “x times y is equal to 16.”2.2 mathematical sentence: a+4≥7verbal sentence: The sum of a number and four is greater than or equal to seven. Now, let us see if you can translate the given mathematical sentence into a verbalsentence.2.3 mathematical sentence: 5 - 2y = -3verbal sentence: ____________________________________________ If your answer is “Five subtracted by twice a number is negative three.” or “Twotimes a number subtracted from five equals negative three.”, then you are correct.2.4 mathematical sentence: 3 (r + 9) < 10verbal sentence: ___________________________________________ If your answer is “Thrice the sum of a number and 9 is less than ten.” or “Threemultiplied by the sum of r and nine is less than ten.”, then you are very good. This meansthat you already know how to translate mathematical sentences into verbal sentences.Let us have a summary of our discussion. To translate a verbal sentence into a mathematical sentence, weuse a symbol that corresponds to every word or phrase in the given verbalsentence until the mathematical sentence is formed To translate a mathematical sentence into a verbal sentence, weuse a word or a phrase to that corresponds to every symbol in the givenmathematical sentence until the thought of the verbal sentence is obtained. 12

Self-check 2A. Translate each verbal sentence into a mathematical sentence. 1. A number added to six is equal to two. 2. A number minus 16 is equal to 38. 3. The difference between 4a and 7 is less than 6. 4. Seven times the sum of 8 and a is grater than or equal to 10. 5. Six times a number y less than four is equal to eight.B. Translate the given mathematical sentence into a verbal sentence. 1. 2x + 5 = 9 2. 9 + 3x = 18 3. 2x – 16 ≤ 4 4. 2(x + 1) = 8 5. 4m – 3 ≥ 16 Answer Key on page 21Lesson 3 Differentiating First-degree Equations from First-degree Inequalities in One Variable In lesson 1, you learned the difference between an equation and inequality. Let usstudy the following.The equations below are first-degree equations in one variable.x=1 5b = 4 2a + 7 = 0 6c – 5 = -2 7 + 8y = 2The equations below are not first-degree equations in one variable.y2 = 1 5b – r = 14 2a + 7b3 = 0 6c4 – 5 = -2 8y = 2x 13

Which of the following equations are first-degree equations in one variable?x + 8 = 15 z3 = 0 3a + 6h = ½ 5r – 1 = 4 t = -7Compare your answers with mine. x + 8 = 15, 5r – 1 = 4 and t = -7Did you get the correct answers? Good!How do you define a first-degree equation in one variable? _______________________________________________________________________________________________You are right. A first-degree equation in one variable is an equation that contains onlyone variable and the variable is raised to exponent 1. Thus, a first-degree equation in x isof the form ax + b = 0 where a is a nonzero real number and b is any real number.Can you give your own examples of first-degree equations in one variable?________________________________________________________________If your examples are equations that contain only one variable and the variable is raised toexponent one, then you are right. Now, let us consider the first set of first-degree equations in one variable that aregiven above.x=1 5b = 4 2a + 7 = 0 6c – 5 = -2 7 + 8y = 2If the symbol = is changed to any of the following relation symbols, <, ≤, > or ≥, then wehave first-degree inequalities in one variable. Some possible results are as follows.x <1 5b > 4 2a + 7 ≤ 0 6c – 5 ≥ -2 7 + 8y < 2What is a first-degree inequality in one variable? ___________________________________________________________________________________________________Correct. A first-degree inequality in one variable is an inequality that contains only onevariable and the variable is raised to exponent 1.A first-degree inequality in x is of the following forms: ax + b < 0 ax + b ≤ 0 ax + b > 0 ax + b ≥ 0where a is a nonzero real number and b is any real number.Give your own examples of a first-degree inequality in one variable?___________________________________________________________________If your examples are of the forms as stated above, then you are correct. 14

Let us have a summary of our discussion. A first-degree equation in one variable is an equation thatcontains only one variable and the variable is raised to exponent 1.A first-degree equation in x is of the formax + b = 0 where a is a nonzero constant and b is any real number. A first-degree inequality in one variable is an inequality thatcontains only one variable and the variable is raised to exponent 1.A first-degree inequality in x is of the following forms:ax + b < 0ax + b ≤ 0 where a is a nonzero real numberax + b > 0 and b is any real number.ax + b ≥ 0 Self-check 3Determine whether each of the following is an example of a first-degree equation or a first-degree inequality in one variable. Explain your answer. 1. x + 1 = 0 2. 9/y2 = 3 3. 4a2 + 4a + 1 > 0 4. m + n = 25 5. 9 = 3c 15

Lesson 4 Applications of Equations and Inequalities Equations and inequalities are used to model some real-life situations. This issuccessfully done by using your knowledge in translating a verbal sentence into an equationor inequality. Let us study the following examples. 1. In 1994, twice the population of a barangay n in Cavite is 50 000. This is modeled by an equation that is obtained by translating the verbal sentence “Twice n is 50 000.” into an equation. Thus, we have 2n = 50 000. 2. The distance d that a vehicle travels is computed by multiplying the rate r by the time t it consumes. In symbols, this is written as d = rt. What equation represents the time consumed by a plane in traveling a distance of 1,468 miles at the rate of 400 mi/hr.? ___________________________ Correct. The equation is 1468 = 400t. 3. Patrick is 4 inches taller than Manny. The sum of their heights is less than 7 feet. Represent this by a first-degree inequality in one variable. ___________________________________________________________ If you use the variable p for Patrick’s height, then Manny’s height is p + 4. (You can also use other variables.) Thus, your final answer must be p + p + 4 < 7. 4. The amount earned by John is three times the amount earned by Armand. What first-degree inequality in one variable will be used to model the situation if you use the variable a to represent the amount earned by Armand and their total earnings is at least P28,000? ___________________________________ You should have represented the amount earned by John as 3a and your final answer must be a + 3a ≥ 28000. 5. Connie’s age is half of Ian’s age. Suppose Ian’s age is represented by i. What first- degree equation in one variable will represent the verbal sentence “Ten years from now, their total ages will be 54.” ? ______________________ 16

Let us check your answer. You should have used ½ i for Connie’s age. Ten years from now, Ian’s and Connie’s ages should be represented by i + 10 and ½ i + 10 respectively. Why? ___________________________ Yes, ten years from now is translated as + 10. Thus, your equation must be i + 10 + ½ i + 10 = 54. Let us summarize what you learned in this lesson. Some real-life situations are modeled by equations or inequalities. To do this, we translate the verbal sentence into an equation or inequality. Self-check 4Read the following situations and do what is required.1. Tom’s weight is 2 lbs less than the weight w of Cherry. Write a first-degree equation in one variable that represents the sentence “The sum of Tom’s and Cherry’s weights is 210 lbs.”2. The perimeter p of a rectangle with length l and width w is given by the formula p = 2l + 2w. The length of the top of a rectangular table is 1 m more than its width. What is the first-degree equation in one variable that relates the perimeter and width of the top of that table if the perimeter is 6 m?3. Jenny sold 20 more magazines than Chris. If you use the variable c to represent the number of magazines sold by Chris, what first-degree equation in one variable represents the sentence “Five times the total number of magazines sold by Jenny and Chris is ten more than seven times the number of magazines sold by Jenny.”?4.The number of P10 coins is 17 decreased by the number of P5 coins. If the variable f is used to represent the number of P5 coins, how will you represent the following: 4.1 number of P10 coins in terms of f 4.2 value of P5 coins 17