MATH 1

Lesson 3 Multiplying Polynomials After having reviewed multiplication of polynomials by a monomial usingdistributive property, you will also apply the same principle when you multiplytwo binomials. Examine theses examples:a. (a + 3) ( a + 2 ) Distributive Property= a ( a + 2) + 3 ( a + 2) Distributive Property= a2 + 2a + 3a + 6 Distributive Property again= a2 + 5a + 6 Adding like termsb. (m + 5)(m – 2) = m(m – 2) + 5(m – 2) Distributive Property = m2 – 2m + 5m – 10 Again Distributive Property = m2 + 3m – 10 Adding Similar Termsc. (3a – 3)(2a + 4) = 2a(3a – 3) + 4(3a – 3) = 6a2 – 6a + 12a – 12 = 6a2 + 6a - 12We can multiply the binomials in another way. ExplorationStudy the following examples: 9

Outside Terms Last Terms Outside First Terms Inside Terms Firsta. ( x + 3 ) ( x + 4) = ( x  x ) + (4  x) + (3  x) + (3  4) Inside OutsideOutside Last TermsFirst Inside Terms Outside Terms First Termsb. (2m5 + m2) ( -m3 + m) = 2m5  (-m3) + 2m5  (m) + m2  (-m3) + m2  mInside Outside You will notice that we associate certain key words in finding the productof terms in the binomials. These words are First Terms, Outside Terms, InsideTerms, and Last Terms. That is why, this is called the FOIL Method for findingthe product of two binomials. Self-check 3A. Find the product using the FOIL method. 1. (a + 2) (a – 6) 2. (6a + 4) (a – 7) 3. (y4 + 3) (y – 6) 4. (3mn + 2n) (m2 + 2mn2) 10

5 .  2 x2 + 1  1 x − 1   3 4  8 6 B. This is a game grid in Algebra. Loop the expressions that complete the multiplication of monomials and polynomials. You can loop it vertically, horizontally and diagonally. One example is illustrated below.x–2 X x – 2x y – 2y -6a + 6 x-y 3 y 3y -3 -9y x + yx+3 2x 2a – 2 -3y + 6y a-2 x2 – y2 4 y+2 8+y -6x – 3y -3 2x + yx–5 4 4x – 20 y+4 9x – 3 y - 5 y2 - y - 20 5x x–3 -8x y–2 -4 2+x x+5 -3 y + 3 x 2x2 + 2x -y 2x x+2 y + 3 y2 + y – 6x+2 -6 5x 2y 2xy + 10y 2y2 + 6y x–y Algebra GridExpression: (2x)(-4) = -8x Answer Key on page 22Lesson Special Product : Difference Between Two Squares You have applied the distributive property and the FOIL method formultiplying binomials. Here are some products applying these two methods. 11

a. (a + 4) (a – 4) = a2 – 4a + 4a – 16 = a2 – 16 b. ( 8- x) ( 8 + x) = 64 + 8x – 8x – x2 = 64 - x2 c. ( 5x + 2) ( 5x – 2) = 25x2 – 10x + 10x – 4 = 25x2 – 4 The above examples show the product of the sum and difference of thetwo terms. How are the first terms of the binomial factors related? Their last terms ? You see that the first terms and the terms of the binomials are the same.They differ only in sign. Now, examine once more the products of the expression we havecomputed above. a2 – 16 64 – x2 25 - x2 How are the first term and second term related to the term in the givenbinomials, in a, b, c, respectively?Remember: Product of ( a + b ) and ( a – b ) The product of the sum and the difference of two terms is equal to the square of the first term minus the square of the second term. That is, ( a + b) ( a – b ) = a2 – b2 The product is called the difference between two squares.Here are other examples:a. ( 3x + 4y) ( 3x – 4y ) = 9x2 – 16 y2b. ( 5 + 8a2 )( 5 – 8 a2 ) = 25 – 64 a4c.  1 x − 1 y  1 x + 1 y  = 1 x4 − 1 y  6 4  6 4  36 16 12

Self-check 4 m m I. Multiply: n 1. ( x + 2) ( x + 2) 2. ( 3m + 4 ) ( 3m – 4 ) 3. (a2 + 7) ( a2 – 7 ) 4. (2r3 + s) (2r3 – s) 5. ( –5t + s) ( –5t – s)II. Consider the rectangle at the right. a.What is the area of the rectangle? b.What is the area of the two small unshaded rectangles? c. What is the difference of the area found in ( a ) and ( b)? d.Find the area of the shaded region. e.How do you compare the areas found in c and d? Answer Key on page 22Lesson 5 Squaring a Binomial We can use paper folding to model multiplying a binomial by itself. Thismeans, we are squaring a binomial. This maybe represented by an expression: (a + b) 2ACTIVITY: Paper Folding A. You can perform this Paper Folding for (a + b)2 Material: one square piece of paper Directions: 13

1. Fold one edge over at a point E to form a vertical crease parallel to the edge. Label the longer and shorter dimension with letter a and b, respectively.2. Fold the upper right-hand corner over onto the crease to locate point F. In folding it (Fig. 2), point F must be of the same distance from the corner as point E.a Eb F FFigure 1 Figure 23. Now fold a horizontal crease through F and label all outside dimensions as in Figure 3 a Ebb F Figure 3 abLook at figure 3. Answer the following questions:a. How many figures were formed from the original one square paper?b. What are these figures? 14

c. Using the labels in Figure 3 for the sides, how will you represent the area of each figure?If you get it right, your representation should look this way.a Ebb ab b2 Fa a2 ab 5. How will you express the sum of the areas of the 4 figures? 6. How will you further simplify a2 + ab + ab + b2 ? So we now can write: a2 + ab + ab + b2 = a2 + 2ab + b2 7. Look back at Figure 1. How will you represent the side of the square? If the side then of the big square is represented by a + b, how will you represent the area of the square? So we write: Area of square = s2 and if S = a + b then A = ( a + b )2 Note that in step 6, we added the areas of the four figures : 1 big square, 1 small square and 2 equal rectangles. The sum of these areas is simply equal to the area of the big squareshown in Figure 1. 15

From this, we can now write: ( a + b ) 2 = a2 + 2ab + b2Examine further these examples:( x + 3 )2 = x2 + 3x + 3x + 9 = x2 + 2 (3x) + 9 = x2 + 6x + 9( y + 4)2 = y2 + 4y + 4y + 16 = y2 + 2(4y) + 16 = y2 + 8y + 16( x – 5)2 = x2 – 5x – 5x + 25 = x2 – 2(5x) + 25 = x2 – 10x + 25( 3x + 4)2 = 9x2 + 12x + 12x + 16 = 9x2 + 2(12x) + 16 = 9x2 + 24x + 16SQUARWEhOatFpAattBeIrNnOdoMyIAoLu see? What we have done is a quick way of squaring binomial. TFhinedstqhueapreroodfuactbuinsoinmgiathl eisdeisqturiabluttoivteheprsoqpueartrye. of the first term, plus orminus twice the product of the two terms, plus the square of the last term.That is:Compare the results.( a + b )2 = a2 + 2 ( ab) + b2First term (1st term) 2 + 2 (1st term 2nd term) + ( 2nd term) 2Second term = a2 + 2ab + b2( a – b )2 = a2 – 2 ( ab) + b2 = a2 – 2 ab + b2When you square a binomial the product is a perfect square trinomial. 16

Here are other examples: 1. ( m + 8 )2 = m2 + 16m + 64 2. ( t – 9 ) 2 = t2 – 18t + 81 3. ( 4x + 1 )2 = 16x2 + 8x + 1 4. ( 7a + b2 )2 = 49a2 + 14ab2 + b4 5. ( 2m2 + 3n)2 = 4m4 + 12m2n + 9n2 6. ( x + y + 3) 2 = [ ( x + y ) + 3 ] 2 = ( x + y)2 + 2[ 3 ( x + y) ] + 32 = x2 + 2xy + y2 + 6x + 6y + 9 Self-check 5 I. Multiply: 1. ( s + 6 )2 2. ( x + 7) 2 3. ( 3m + 2)2 4. ( r – 10 ) 2 5. ( 2a – 9b ) 2 6.  1 a + 1 2 3 2 7.  x + 2 2  3 8. ( a + b+ 4)2 II. Solve the following problems: 17

1. A square has a side of 2x + 6 units. What expression dddd represents the area of the square?2. Find the area of the shaded region. x2 + 3x2 + 1 x2 + 3 x2 + 1 Answer Key on page 22 Let’s summarize To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial, applying the rules of exponents for multiplication. In multiplying a polynomial by another polynomial, each term in the multiplier is multiplied by each of the terms in the multiplicand. Similar terms in the partial products are combined. In applying the distributive property of multiplication, the FOIL (First, Outer, Inner, Last) method could guide you in finding the product. There is a shorter way of multiplying binomials. The results are special products. Two of these cases are the product of the sum and difference of two terms and the square of a binomial. The product of the sum and difference of two binomials is equal to the square of the first term minus the square of the second term. The square of a binomial is equal to the square of the first term plus or minus twice the product of the first term and second term plus the square of the last term. 18

What to do after (Posttest)I. Multiple Choice. Choose the letter of the correct answer.1. The product of 2a and (3b2 + 3b – 2) isa. 6ab + 6b – 4a c. 6b2 + 6b – 4ab. 6ab2 + 6ab – 4a d. 5ab2 + 5ab – 4a2. Multiplying the binomials, (a + 2)(a – 6) gives the middle term,a. 2a c. – 4ab. 6a d. + 4a3. The product of the sum and difference of two binomials is 100 a2b4 – 81. The first term of each binomial isa. 50 abb. 10 abc. 50 ab2d. 10 ab24. When you multiply these two polynomials ( 1 a + 2 ) (1 a – 2 ), the product is 5 35 3a. 1 a2 4 c. 2 a2 – 4 – 59 10 9 d. 1 a2 – 4b. 1 a2 25 25 95. The square of (2x + 4) isa. 4x2 + 4 + 16 c. 4x2 + 16x + 16b. 4x2 + 4 + 16x d. 8x2 + 8x + 8II. Multiply:a. 2x2 ( 4x2 – 2x + 6) 19

b. 3a ( 4a + 10)c. ( x + 8) ( x + 5)d. ( 2x + 3) ( x – 4 )e.  1 m + 1   1 m - 1   4 2   3 2 III. Find the special product of the given polynomials.a. ( x + 1 ) ( x – 1 )b. ( r + 8 ) ( r – 8 )c. ( x + 1 ) ( x – 1 ) 22d. (m3n – 5n ) ( m3n + 5n )e. ( t – 3 )2f. (2x – 1 )2g. ( b + 2 ) 2 3IV. In the following, identify which are true and which are false. If incorrect, write the correct answer.a. ( m + n )2 = m2 + n2b.  x + 1 2 = x2 − 1 x + 1  4 8 16c. ( 3x + 5 ) 2 = 9x2 + 30x + 25d.  2 a2 − b 2 a2 + b = 4 a4 − b2  3  3  9V. Answer: a. What is the relationship between a – b and b – a. b. Find ( a – b )2 and ( b – a )2. Make a statement from the results. Answer Key on page 23 20

Key to CorrectionPre Test page 3 I. 1. d 2. d 3. d 4. b 5. dII. 1. 6m2 + 10m 2. 2a4 + 3a3 – 4a2 3. l2 + 9l + 18 4. 5x2 – 14x – 3 5. 1 x2 + 1 x − 1 10 10 9III. 1. y2 - 36 2. w2 – 49 3. y2 − 9 16 4. s4 – 9t2 5. m2 – 16m + 64 6. 4n2 – 4n + 1 7. a2 + 4 a + 41 7 49IV. a. No. Because ( x + y)2 = x2 + 2xy + y2 b. ( x + 4) 2 = x2 + 8x + 16Lesson 1 Self- Check 1 page 6I. 1. y8 2. – y14 3. a9 4. -24x7 5. 8 b3 6. 25a4 b2 7. 64 8. 16x6II. a. A = 25x2 b. V = 27a3Lesson 2 Self- Check 2 page 8 21

A. 1. b 2. d 3. f 4. c 5. aB. 1. 8x2 – 24x 2. – 10m3 + 5m2 3. – 6y9 + 6y8 + 3y7 – 12y6 4. 6m5 -5m4 +16m3 5. .18x4 +2.4x3 + .05xC. a. A = 21m2 -15m b. A = 20x2 + 40xLesson 3 Self- Check 3 page 11 A. 1. a2 -4a – 12 2. 6a2 -44a -28 3. y5 – 6y4 +3y - 8 4. 3m3n + 6m2n3 + 2m2n + 4mn3 5. 1 x3 − 1 x2 + 1 x + 1 12 9 32 24Lesson 4 Self- Check 4 page 13 1. x2 + 4x + 4 2. 9 m2 – 19 3. a4 – 49 4. 4r6 – s2 5. 25t2 – s2Lesson 5 Self- Check 5 page 17 1. s2 + 12s + 36 2. x2 + 14s +49 3. 9m2 +12m +4 4. r2 –20r + 100 5. 4a2 – 36 ab + 81 b2 6. 1 a2 + 1 a + 1 9 34 7. x2 + 4 x + 4 39 8. a2 + b2 + 2ab + 8a + 8b + 16 22

Posttest page 19I. I, b 2. c 3. d 4. d 5. cII. 1. 8x4 – 4x3 + 12x2 2. 12a2 + 30a 3. x2 + 13x + 40 4. 2x2 – 5x -12 5. 1 m2 + 1 m − 1 12 24 4III. 1. x2-1 2. r2 – 64 3. x2 − 1 44. m6n2 – 25n25. t2 -6t +96. 4x2 – 4x +17. b2 + 4 b + 4 39IV. 1. False, ( m+n)2 = m2 + 2mn + n22. False,  x + 1 2 = x2 + 1 x + 1  4 2 163. True4. TrueV. a. ( a - b) and ( b - a) are additive inversesb. (a - b)2 = a2 - 2ab + b2 and ( b - a)2 = b2 – 2ab + a2c. The squares of two quantities which are additive inverses of each other are equal. END OF MODULE 23

Module 17 The XYZ FactorsWhat this module is all about This module is about factoring polynomials. This deals on products of polynomialswhose terms have a common monomial factor, trinomials which are products of twobinomials, trinomials which are squares of a binomial, and products of the sum anddifference of two quantities. From time to time, you will be referred to the lessons in thepreceding module on special product because special product and factoring are reverseprocesses. You will see also how factoring can be applied widely in many mathematicsproblems.This module includes the following lessons:Lesson 1 Factoring and Common FactorsLesson 2 Factoring Expressions of the Form x2 + bx + cLesson 3 Factoring Expressions of the Form ax2 + bx +cLesson 4 Factoring Perfect Square TrinomialLesson 5 Factoring the Difference Between Two SquaresLesson 6 Sum and Difference of Two Cubes What you are expected to learn After working on this module you should able to: Factor polynomials in each of the following cases:  polynomials whose terms have a common monomial factor  trinomials which are products of binomials  trinomials which are squares of a binomial  difference of two squares  sum and the difference between two cubes 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! 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)I. Multiple Choice. Choose the letter of the correct answer.1. If one of the factors of 18x5y + 12x4y – 6x3y + 3x2y is 3x2y, what is the other factor?a. 6x4 + 4x3 – 2x2 + x c. 6 + 4x – 2x2 + x3b. 6x3 + 4x2 – 2x + 1 d. 6x + 4x2 – 2x3 + x42. Which of the following expressions has (x – 4) and (x + 3) as factors?a. x2 + 7x + 12 c. x2 – 7x – 12b. x2 + x – 12 d. x2 – x – 123. Which of the following is TRUE?a. 25x2 – 120xy + 144y2 = (5x – 12y)2b. 9a2 – 24a – 16 = (3a – 4)2c. 16x2 + 12xy + 9y2 = (4x + 3y)2d. x2 – y2 = (x – y)24. Which of the following is a factor of 9a2 – 25b2? i. 3a – 5b ii. 3a + 5ba. i c. i and iib. ii d. cannot be determined5. Which of the following are factors of a3 – 8b3?a. a – 2b; a2 – 4ab + b2b. a – 2b; a2 – 2ab + 4b2c. a – 2b; a2 + 4ab + 4b2d. a – 2b; a2 + 2ab + 4b2II. Determine the factors of the following polynomials.A. 1. 3x+ 12 2. 10m + 6m2 3

3. xy2z3 – x2y3z4 4. ay + by – cy + 3y 5. x2 – 49 6. 1 a2 − 9 b2 4 16 7. x4 – y4 8. (a – b)2 – 16B. 1. a2 + 14a + 49 2. m2 + 6n + 8 3. t2 – 15t + 16 4. x6 – 27 5. 8a3 + 125 6. 1 r6 − 8 27 7. 2x2 – 5x – 12 8. 4n2 – 15n + 9C. Answer these questions: a. If the area of a square is 25x2 + 30x + 9, what is the measure of its side? b. A rectangle has an area of x2 + 12x + 32. If the length is represented by x +8, what is the width of the rectangle? Answer Key on page What you will doLesson I Factoring and Common Factor In the previous module, you studied about special ways of finding the productof polynomials. The results are called special products. This time, we will do the otherway around. You will be given the product and you will determine its factors. This process 4

is factoring. It is the reverse process of multiplying. When we factor expressions, wedetermine an equivalent expression that is a product of two or more expressions. Before we proceed further, let us review what you have learned aboutfactors, prime factorization and greatest common factor. Following are some examples that will enable you to recall some conceptsyou have studied before.1. Factor: a) 36 b) 54 c) 72Solution: For each given number, we may write the factorization process in a form of factor tree as shown here:36 54 729x 4 9x 6 9x83 x 3x2 x 2 3 x 3X 2 x 3 3 x 3x 2 x 2X2 Look at the darkened numbers in the last row. They are the primefactorization of 36, 54 and 72. Why do you think we call these the prime factorization of the given number? We say that if a given number is written as a product of prime factors, theyare factored completely. From the prime factorization as shown , we can determine the GreatestCommon Factor (GCF) of the numbers 36, 54, and 72.36 = 3 x 3 x 2 x 254 = 3 x 3 x 2 x372 = 3 x 3 x 2 x 2 x2GCF = 3 x 3 x 2 = 18 We say that 18 is the largest number that could divide 36, 54 and 72 exactly.It is also called their greatest common divisor.The concept of GCF can now be extended to monomials and polynomials. 5

Study the following examples: 1. Find the GCF of 4x2, 8x4 and 12x6 SOLUTION: We factor each of the given monomials where the highest common factor can be determined among the three monomials. 4x2 = 4x2  1 8x4 = 4x2  2x2 12x 6 = 4x2  3x4 GCF = 4x2The greatest common factor of 4x2, 8x4, and 12x6 is 4x2. 2. Find the GCF of 8m3n4 – 20m2n3 -10mn2 8m3n4 = 2mn2 x 4m2n2 -20m2n3 = 2mn2 x -10mn -10mn2 = 2mn2 x - 5 = 2mn2 GCF The processes of prime factorization and finding the GCF will be very usefulin solving problems involving factoring.Remember: Factoring is the process of obtaining equivalent expression thatis the product of two or more expressions.A. Common Monomial Factor Suppose you are given a rectangle whose area is represented by10x2 +15xy, with a width of 5x, how do we express the length of this rectangle? A situation like this can be solved with the knowledge of common monomialfactor. Study the illustrative examples below on factoring polynomials whoseterms have a common monomial factor. 6