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9M02_Polynomials_Avanti Module

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9M02 – Polynomials 14 Fifth Edition 9M02. Polynomials TABLE OF CONTENTS 9M02. Polynomials 14 9M02.1 Introduction to Polynomials ...............................................................................................................................................15 9M02.2 Zeroes of a Polynomial...........................................................................................................................................................17 9M02.3 Remainder Theorem...............................................................................................................................................................19 9M02.4 Factor Theorem.........................................................................................................................................................................21 9M02.5 Algebraic Identities ................................................................................................................................................................. 23 Advanced Practice Problems.................................................................................................................................................................26 Olympiad Practice Problems.................................................................................................................................................................27

9M02 – Polynomials 15 9M02.1 Introduction to Polynomials CONCEPTS COVERED 1. Algebraic expressionss 2. Polynomials 3. Degree of a polynomial IN CLASS EXERCISE LEVEL 1 Q1. The standard form of a linear polynomial is B) ������������ D) Both A and B A) ������������ + ������ C) ������������3 Q2. Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials: (A biquadratic polynomial has a degree which is twice that of a quadratic polynomial) I. ������ + ������2 + 4 II. 3������ − 2 III. 2������ + ������2 IV. 3������ V. ������2 + 1 VI. 7������4 + 4������3 + 3������ − 2 Q3. Find the coefficient of ������ in the following polynomials and classify them as monomials, binomials and trinomials. I. ������2 + ������ II. ������100 + ������50 + 1 IV. 2376 6 III. ������������ + ������ V. ������������2 + ������������ + ������������ LEVEL 2 Q4. Give an example of: II. Monomial of degree 12 I. Trinomial of degree 99 Q5. [PSV] Which of the following are polynomials in one variable? Give reasons for your answer. Also, write the degree of each of the polynomials. I. 3√������ + 7 II. ������100 + 101 III. ������ + √69 IV. ������5 + 5 V. ������2 + ������5 VII. ������ + 2������ + 3������ ������ VI. 2√������ + ������2 VIII.������2������ + 9������2 + 17������������3 Q6. Fill in the blanks: I. If the radius of a circle is ������, then its circumference is given by the polynomial _______ and its area is given by the polynomial ______. II. If a polynomial depends on variable ������, such that it is 6 more than twice the cube of variable ������, then the polynomial is given by _________. Q7. Write down a quadratic monomial and a linear binomial. I. What is the degree of the polynomial obtained on multiplying the two polynomials you have written? II. Would the result be the same, if a quadratic trinomial is multiplied with a linear binomial? Hence, what would be the degree of a polynomial obtained on multiplying two polynomials of degrees 4 and 13? 9M02.1

9M02 – Polynomials 16 HOMEWORK LEVEL 1 Q1. Write the standard form of a cubic polynomial with real coefficients. Q2. Identify constant, linear, quadratic and cubic polynomials from the following polynomials: I. ������(������) = 3 II. ������(������) = 2������3 − 7������ + 4 III. ℎ(������) = −3������ + 1 IV. ������(������) = 2������2 − ������ + 4 2 VI. ������(������) = 3������3 + 4������2 + 5������ − 7 V. ������(������) = 4������ + 3 Q3. Which of the following is not a quadratic polynomial? A) ������2 + 5������ + 6 B) ������2 − 5������ + 6 C) 1 + (������2 − 2������) D) (������ − 2)(������ + 2) − (������2 + 5������) Q4. Find the coefficient of ������2 in the following polynomials. Also, find number of terms in each polynomial. I. ������������2 + ������������ + ������ II. 1769 III. 6������4 + ������ + 2 + 5������ IV. 5������4 − 12������2 + 3������ + 4������3 − 90 + ������2 LEVEL 2 Q5. Give an example of II. Monomial of degree 3 I. Trinomial of degree 1 Q6. Is √11 a polynomial? If yes, then find its degree and number of terms. Q7. Which of the following are polynomials in one variable? Give reasons for your answer. Also, write the degree of each of the polynomials. 1 II. ������4 + ������2 + 3������ + 18 IV. ������������, where ������ is a natural number. I. ������3 + ������2 VI. ������������ + ������������������ + ������5������ III. ������ + 1 ������2 V. 9������ + √������ Q8. If the side of a square is ������, then its perimeter is given by the polynomial _______ and its area is given by the polynomial ______. Q9. Two polynomials in variable ������, ������������3 + ������������2 + ������������ + ������ and ������������3 + ������������2 + ������������ + ������, are: I. Added II. Multiplied III. Subtracted What is the maximum possible degree of resultant polynomial in these cases? PUZZLE What was your answer? What was your partner’s answer? Can you think of a reason why your answers came out to be the same? 9M02.1

9M02 – Polynomials 17 9M02.2 Zeroes of a Polynomial CONCEPTS COVERED 1. Value of a polynomial 2. Zeroes of a polynomial IN CLASS EXERCISE LEVEL 1 II. ������ = −3 IV. ������ = 0.5 Q1. Find the value of polynomial 3������3 − 5������2 + ������ − 1 at: I. ������ = −2 III. ������ = 0 V. ������ = 1 Q2. Verify whether the following are zeroes of the polynomial, indicated against them. I. ������2 + ������ − 6 ; ������ = 1 II. ������ + ������ ; ������ = −������ IV. ������4 − 16; ������ = −2 III. 9������ − 27 ; ������ = 3 V. ������ − 1; ������ = 5 5 Q3. If ������ = 4 is a zero of the polynomial ������(������) = 6������3 − 11������2 + ������������ − 20, find the value of ������. 3 Q4. Find zeroes of the following polynomials: I. ������������ + ������ ; ������ ≠ 0 II. ������ − 0.79 IV. 7������ − 77 III. 2������ + 9 V. ������ + 1 2 LEVEL 2 Q5. If ������ = 0 is a zero of the polynomial ������(������) = 2������3 − 5������2 + ������������ + ������, then find the value of ������. Now, if ������ = 2 is also a zero of this polynomial, then find the value of ������. Q6. What should be added to the polynomial ������2 − 5������ + 4, so that 3 is a zero of the resulting polynomial? HOMEWORK LEVEL 1 Q1. Number of zeroes of a non-zero constant polynomial is _______________. Q2. Number of zeroes of a zero polynomial ______________. Q3. If ������(������) = ������2 − 2√2������ + 1, then find ������(3√2). Q4. Find ������(0) and ������(1) for the following polynomials: II. ������(������) = 5������ + ������2 IV. ������(������) = 4������101 − 1001������2015 I. ������(������) = ������������3 + 8 III. ������(������) = (������ − 1)(������ + 1) V. ������(������) = 100������499 − 99������500 9M02.2

9M02 – Polynomials 18 Q5. Verify whether the following are zeroes of the polynomial, indicated against them. I. 3������3 + ������2 − 2������ − 7 ; ������ = 2 II. ������2 − 16 ; ������ = −4 III. 7������4 − 4������3 ; ������ = 0 IV. 2������ − 3; ������ = 1.5 V. ������2 + 4������ + 1; ������ = −1 Q6. For what value of ������, is −4 a zero of the polynomial ������2 − ������ − (2������ + 2)? Q7. Find zeroes of the following polynomials: I. ������������, ������ ≠ 0 II. 5������ + 3 III. 9������ + 72 2 V. 3������ + 8 IV. ������ − 5 LEVEL 2 Q8. If ������, ������, 8 and −1 are zeroes of the polynomial ������(������), then find I. ������(������) − ������(������) II. (������(������))2 − ������(−1) III. ������(−1)−������(������)+������+9 ������(8)+������(������)+501 Q9. If ������ = −2 is the zero of the polynomial √2 (������ + ������) and is also a zero of the polynomial ������������2 + ������������ + 2√2, then find the value of ������. ADVANCED QUESTIONS Q1. [PSV] Find zeroes of the following polynomials: I. (������ − 1)(������ + 2)(������ − 4) II. ������2 − 4 III. ������2 + 6������ + 9 IV. 16������2 − 8������ + 1 V. 81������2 − 36 VI. ������3 Q2. What can be the expression for a cubic polynomial, if 2, 3 and 4 are its zeroes? 9M02.2

9M02 – Polynomials 19 9M02.3 Remainder Theorem CONCEPTS COVERED 1. Division of polynomials 2. Remainder theorem IN CLASS EXERCISE LEVEL 1 Q1. Solve the following using long division method. II. (2������3 + ������2 − ������ + 4) ÷ (������2 + 2������ − 2) IV. (5������2 − 3) ÷ (������3 − 5) I. (3������3 + 2������2 + ������ + 1) ÷ (������ + 1) III. (������2 − ������ + 1) ÷ (2������ + 1) Q2. In the following problems, what will be the remainder? Solve using remainder theorem. I. (������2 + 1) ÷ (������ − 1) II. (3������3 + ������2 − ������ − 9) ÷ (3������ − 2) III. (4������3 + 3������2 + 2������ + 1) ÷ (������ + 5) Q3. Find the remainder when ������(������) = 4������3 − 12������2 + 14������ − 3 is divided by ������(������) = ������ − 1. 2 LEVEL 2 Q4. Check whether the polynomial ������(������) = 4������3 + 4������2 − ������ − 1 is a multiple of 2������ + 1. Q5. The polynomial ������(������) = ������������3 + 9������2 + 4������ − 8 when divided by (������ + 3) leaves a remainder 10 (1 − ������). Find the value of ������. Q6. On dividing ������(������) = ������4 − 2������3 + 3������2 − ������������ + 8 by (������ − 1), we get a remainder of 5. Find the remainder when ������(������) is divided by (������ − 2). Q7. Let ������1 and ������2 be the remainders when the polynomials ������3 + 2������2 − 5������������ − 7 and ������3 + ������������2 − 12������ + 6 are divided by (������ + 1) and (������ − 2) respectively. If 2������1 + ������2 = 6, find the value of ������. HOMEWORK LEVEL 1 Q1. Solve the following using long division method. II. (5������4 − ������2 − 9) ÷ (������3 − 3������ + 2) IV. (9������3 − ������ + 1) ÷ (������2 − 1) I. (4������2 − ������ + 9) ÷ (4������ − 7) III. (������3 − 1) ÷ (������ − 1) Q2. In the following problems, what will be the remainder? Solve using remainder theorem. I. (������������2 + ������������ + ������ ) ÷ (������ + 7) II. (������3 − ������ + 1) ÷ (������ + 3) Q3. When a polynomial of degree 3 is divided by another polynomial, what will be the maximum possible degree of the remainder? Give reasons for your answer. Q4. Find the remainder when ������(������) = ������3 − ������������2 + 6������ − ������ is divided by (������ − ������). LEVEL 2 Q5. State and prove the remainder theorem. 9M02.3

9M02 – Polynomials 20 Q6. Check whether the polynomial ������(������) = 2������ + 3 is a factor of the polynomial ������(������) = 4������3 + 4������2 + ������ + 6. Q7. If the polynomials ������������3 + 4������2 + 3������ − 4 and ������3 − 4������ + ������ leave the same remainder when divided by (������ − 3), find the value of ������. Q8. The polynomial ������(������) = 2������3 − 3������2 + ������������ − 3������ + 9 when divided by ������ + 1, leaves the remainder 16. Find the value of ������. Also, find the remainder when ������(������) is divided by ������ + 2. Q9. The polynomials ������������3 + 3������2 − 3 and 2������3 − 5������ + ������ when divided by ������ − 4 leave remainders ������ and ������ respectively. Find the value of ������, if 2������ = ������. 9M02.3

9M02 – Polynomials 21 9M02.4 Factor Theorem CONCEPTS COVERED 1. Factor theorem 2. Factorisation using middle term splitting method 3. Factorisation using factor theorem IN CLASS EXERCISE LEVEL 1 Q1. Show that (������ − 1) is a factor of ������10 − 1 and also of ������11 − 1. Q2. For what values of ������ is 2������3 + ������������2 + 11������ + ������ + 3 exactly divisible by (2������ − 1)? Q3. Use the factor theorem to determine whether ������(������) is a factor of ������(������) in each of the following cases: I. ������(������) = ������3 + 3������2 + 3������ + 1, ������(������) = ������ + 2 II. ������(������) = ������3 − 4������2 + ������ + 6, ������(������) = ������ − 3 Q4. Factorise the following by splitting the middle term: I. ������2 + 3������ + 2 II. 4������2 − 14������ + 6 III. 5������2 + 11������ − 12 LEVEL 2 Q5. Without actual division, prove that 2������4 − 6������3 + 3������2 + 3������ − 2 is exactly divisible by ������2 − 3������ + 2. Q6. Factorise: 2������4 + ������3 − 14������2 − 19������ − 6. Q7. Without using long division method, factorise the polynomial ������3 − 6������2 + 11������ − 6. HOMEWORK LEVEL 1 Q1. Determine the value of ������ for which the polynomial 2������4 − ������������3 + 4������2 + 2������ + 1 is divisible by 1 − 2������. Q2. Find the value of ������, if ������ − 1 is a factor of ������(������) in each of the following cases: I. ������(������) = 2������2 + ������������ + √2 II. ������(������) = ������������2 − √2������ + 1 Q3. Factorise by splitting the middle term. II. 3������2 + 10������ − 25 I. 6������2 − 19������ + 8 III. 2������2 + 3������ − 14 LEVEL 2 Q4. Without actual division, prove that 2������4 − 5������3 + 2������2 − ������ + 2 is exactly divisible by ������2 − 3������ + 2. Q5. Without actual division, prove that ������4 + 2������3 − 2������2 + 2������ − 3 is exactly divisible by ������2 + 2������ − 3. Q6. Factorise 9������3 − 27������2 − 100������ + 300, if it is given that (3������ + 10) is a factor of it. Q7. Show that (������ − 2) is a factor of the polynomial ������(������) = 2������3 − 3������2 − 17������ + 30 and hence, factorise ������(������). Q8. If both ������ − 2 and ������ − 1 are factors of ������������2 + 5������ + ������, show that ������ = ������. 2 Q9. Use factor theorem to verify that ������ + ������ is a factor of ������������ + ������������ for any odd positive integer ������. Q10. Without using long division method, factorise the polynomial ������4 + ������3 − 7������2 − ������ + 6. 9M02.4

9M02 – Polynomials 22 ADVANCED QUESTIONS Q1. If ������2 − 1 is a factor of ������������4 + ������������3 + ������������2 + ������������ + ������, show that ������ + ������ + ������ = ������ + ������ = 0 Q2. Find the values of ������ and ������ so that the polynomial ������3 − ������������2 − 13������ + ������ has (������ − 1) and (������ + 3) as factors. Q3. If ������������3 + ������������2 + ������ − 6 has ������ + 2 as a factor and leaves a remainder 4 when divided by (������ − 2), find the values of ������ and ������. 9M02.4

9M02 – Polynomials 23 9M02.5 Algebraic Identities CONCEPTS COVERED 1. Expansion and factorisation of algebraic expressions 2. Algebraic identities IN CLASS EXERCISE LEVEL 1 Q1. Use suitable identities to find the following products: II. (������2 + 3) (������2 − 3) I. (3������ + 4)(3������ − 5) III. (−2������ + 5������ − 3������)2 22 Q2. Factorise the following using appropriate identities: IV. [3 ������ + 3 I. 9������2 + 6������������ + ������2 2 II. 2������2 + ������2 + 8������2 − 2√2������������ + 4√2������������ − 8������������ 1] Q3. If ������ + ������ + ������ = 0, show that ������3 + ������3 + ������3 = 3������������������ LEVEL 2 Q4. Give possible expressions for the length and breadth of a rectangle whose area is given by the expression: 25������2 − 35������ + 12. Q5. Verify that ������3 + ������3 + ������3 − 3������������������ = 1 (������ + ������ + ������)[(������ − ������)2 + (������ − ������)2 + (������ − ������)2] 2 Q6. Simplify: (������ − 2 3 − (������ + 2 3 ������) ������) 33 Q7. Verify that: ������������ [(������ + ������) (1 + 1) − 4] = (������ − ������)2 ������ ������ Q8. If ������ + 1 = 7, then find the value of ������ 3 + ������13. ������ Q9. Using suitable identity, find the value of 872−88773×+1133+3 132. Q10. If ������ + 2������ = 10 and ������������ = 15, find ������3 + 8������3. HOMEWORK LEVEL 1 Q1. Expand each of the following using suitable identities: I. (1 ������ − 1 ������ + 2 II. (������ − 1 3 42 1) 2 ������) Q2. Factorise the following using appropriate identities: I. 4������2 + 9������2 + 16������2 + 12������������ − 24������������ − 16������������ II. ������2 − ������2 100 9M02.5

9M02 – Polynomials 24 Q3. Verify: II. ������3 − ������3 = (������ − ������)(������2 + ������������ + ������2) I. ������3 + ������3 = (������ + ������)(������2 − ������������ + ������2) Q4. Simplify: (������ + ������ + ������)2 + (������ − ������ − ������)2. LEVEL 2 Q5. What are the possible expressions for the dimensions of the cuboids whose volumes are given below? I. 3������2 − 12������ II. 3������������2 + 2������������ − 5������ Q6. Without actually calculating the cubes, find the value of each of the following: I. (−12)3 + (7)3 + (5)3 II. (28)3 + (−15)3 + (−13)3 Q7. Factorise: ������7 + ������������6. Q8. If ������ and ������ are two positive real numbers such that ������2 + 4������2 = 17 and ������������ = 2, then find the value of (������ + 2������). Q9. If ������ and ������ are two positive real numbers such that 8������3 + 27������3 = 730 and 2������2������ + 3������������2 = 15, then find the value of 2������ + 3������. Q10. Prove that: (������ + ������)3 + (������ + ������)3 + (������ + ������)3 − 3(������ + ������)(������ + ������)(������ + ������) = 2(������3 + ������3 + ������3 − 3������������������). POLYNOMIAL PUZZLE: 4 −2 −8 5 −3 −15 20 6 120 What is the structure of this puzzle? The puzzle is structured so that the four numbers on the top left corner are multiplied in rows and columns with the products in the right hand column and bottom row. This is illustrated as below. 4 −2 −8 4 × −2 = −8 5 −3 −15 5 × −3 = −15 20 6 120 20 × 6 = 120 4 × 5 = 20 −2 × −3 = 6 −8 × −15 = 120 Now solve the following puzzles. Q1. 5 −7 10 2 Q2. 3 −15 −2 −240 9M02.5

9M02 – Polynomials 25 Q3. ������ − 3 −2������ + 6 3 −5������ + 1 30������2 − 96������ + 18 Q4. −4 2������ − 6 −8 −8������2 + 72 Q5. ������ + 3 2 8������ 12������ Q6. ������ + 3 2������ + 10 2������ + 6 7 Q7. 2������ + ������ 2������ + ������ 2������ + ������ ������2 − 9������2 8������3 + ������3 + 12������2������ + 6������������2 Q8. ������ + 3������ ������2 + 9������2 + 6������������ Q9. ������4 + 81������4 − 18������2������2 ������ − ������ ������2 − ������2 ������2 + ������2 + ������������ ������4 + ������4 + ������2������2 9M02.5

9M02 – Polynomials 26 Advanced Practice Problems Q1. What must be subtracted from 4������4 − 2������3 − 6������2 + ������ − 5 so that the result is exactly divisible by 2������2 + ������ − 1? Q2. If ������2 + ������2 + ������2 − ������������ − ������������ − ������������ = 0, then the value of ������: ������: ������ is A) 1: 1: 1 B) 1: 2: 3 C) 1: 3: 4 D) 1: 4: 1 Q3. The value of (1 + 1) (1 + 1 ) (1 + 1 ) (1 + 1 ) (1 + 1 ) is ������ ������−1 ������−2 ������−3 ������−4 A) ������−3 B) ������+1 C) ������+1 D) ������−1 ������ ������−4 ������−1 ������−4 Q4. (1 − 1) (1 − 1 ) (1 − 1 ) … (1 − 1 ) is D) 2������ ������ ������+1 ������+2 ������+������ ������−1 A) 1 B) 1 C) ������−1 ������ 2������ 2������ Q5. If ������ = 999, then the value of ������(������2 + 3������ + 3) is A) 999 999 B) 9 999 999 C) 999 999 999 D) None of these Q6. If ������ + ������ = 1, then what is the value of ������3 + ������3? ������ ������ Q7. If ������(2 − √3) = ������(2 + √3) = 1, then what is the value of ������2 − ������2? Q8. If ������������ = ������, ������������ = ������ and ������������ = ������, then what is the value of ������������������? A.P.P.

27 9M02 – Polynomials – Olympiad Practice Problems INSTRUCTIONS Answer each of the following questions. 1. If ‫ݔ‬ଶ + ‫ ݕݔ‬+ ‫ = ݔ‬12 and ‫ݕ‬ଶ + ‫ ݕݔ‬+ ‫ = ݕ‬18, then the value of ‫ ݔ‬+ ‫ ݕ‬is ? A) 5 or െ6 B) 3 or 4 C) 5 or 3 D) 6 or െ3 2. ቂଵ + ଵ + ଶ + ସ + ଵା଼௫ఴቃ is equal to? ଵା௫ ଵା௫మ ଵା௫ర ଵି௫ A) 1 B) 0 C) ଼ D) ଵ଺ ଵି௫ఴ ଵି௫భల 3. Simplify : ௔రି௔య௕ି௔௕యା௕ర ௔రା௔య௕ି௔௕యି௕ర A) (௔ି௕)మ B) ௔మି௕మ C) ௔ି௕ D) 1 ௔మା௕మ ௔ା௕ ௔ା௕ 4. Factors of (ܽଶ + ܽ)ଶ + 4(ܽଶ + ܽ) െ 12 are A) (ܽଶ + ܽ + 6)(ܽ + 2)(ܽ െ 1) B) (ܽଶ െ ܽ + 6)(ܽ െ 2)(ܽ + 1) C) (ܽଶ + ܽ + 6)(ܽ െ 2)(ܽ െ 1) D) (ܽଶ + ܽ + 6)(ܽ + 2)(ܽ + 1) 5. Find the square root of (4ܽ + 5ܾ + 5ܿ)ଶ െ (5ܽ + 4ܾ + 4ܿ)ଶ + 9ܽଶ A) ξ3(ܾ + ܿ) B) 3(ܾ + ܿ െ ܽ) C) 3(ܾ + ܿ) D) 3(ܾ + ܿ െ ܽ) D) All of these. 6. One of the dimensions of the cuboid whose volume is 16‫ݔ‬ଶ െ 26‫ ݔ‬+ 10 is A) 2 B) (8‫ ݔ‬െ 5) C) (‫ ݔ‬െ 1) 7. (௔ି௕)యି(௔ା௕)య + ܽ(ܽଶ + 3ܾଶ) = _________ ? ଶ A) ܽଷ െ ܾଷ B) (ܽ + ܾ)ଷ C) ܽଷ + ܾଷ D) (ܽ െ ܾ)ଷ 8. If ξ4‫ݔ‬ସ + 12‫ݔ‬ଷ + 25‫ݔ‬ଶ + 24‫ ݔ‬+ 16 = ܽ‫ݔ‬ଶ + ܾ‫ ݔ‬+ ܿ if ‘ܽ, ܾ, ܿ > 0’ then which of the following is true? A) 2ܾ = ܽ െ ܿ B) 2ܽ = ܾ + ܿ C) 2ܾ = ܽ + ܿ D) 2ܾ = ܿ െ ܽ 9. A polynomial function is defined to be ݂(‫ ݔ( = )ݔ‬+ 3)(‫ ݔ‬െ 2)ଶ which choice describe the graph’s behavior at the ‫ݔ‬- axis? A) The graph only touches the ‫ݔ‬-axis at ‫ = ݔ‬െ3, but crosses the ‫ݔ‬-axis at ‫ = ݔ‬2. B) Touches at ‫ = ݔ‬3, but crosses at ‫ = ݔ‬െ2 C) Touches at ‫ = ݔ‬2, but crosses at ‫ = ݔ‬െ3 D) Touches at ‫ = ݔ‬െ2, but crosses at ‫ = ݔ‬3 10. ܳ(‫ݔ‬, ‫ݔ = )ݕ‬ଷ െ 3‫ݔ‬ଶ‫ ݕ‬+ ‫݌‬ଵ‫ݕݔ‬ଶ + ‫݌‬ଶ‫ݕ‬ଷ (‫ ݔ‬െ ‫ )ݕ‬and (‫ ݕ‬െ 2‫ )ݔ‬are two factors of the expression ܳ(‫ݔ‬, ‫)ݕ‬, then find the values of ‫݌‬ଵ and ‫݌‬ଶ? A) ଵ and ଷ B) ଷ and ଵ ଶଶ ଶଶ C) െ1 and 3 D) ‘଻ and ିଷ’ ଶଶ 11. Given the graph below, which choice lists possible factors of this polynomial function? A) (‫ ݔ‬െ 4)(‫ ݔ‬െ 1)(‫ ݔ‬+ 1) B) ‫ = ݔ‬െ4; ‫ = ݔ‬െ1; ‫ = ݔ‬1 C) ‫ = ݔ‬4; ‫ = ݔ‬1; ‫ = ݔ‬െ1 D) (‫ ݔ‬+ 4)(‫ ݔ‬+ 1)(‫ ݔ‬െ 1) 12. If ܽଶ + ܾଶ + ܿଶ െ ܾܽ െ ܾܿ െ ܿܽ = 0; then the value of ܽ ‫ ܿ ׷ ܾ ׷‬is A) 1 ‫ ׷‬1 ‫ ׷‬1 B) 1 ‫ ׷‬2 ‫ ׷‬3 C) 1 ‫ ׷‬3 ‫ ׷‬4 D) 1 ‫ ׷‬4 ‫ ׷‬1 Avanti – 9M02 – Polynomials – Olympiad Practice Problems ss

13. The value of ቀ1 + ଵቁ ቀ1 + ଵ ቁ ቀ1 + ଵ ቁ ቀ1 + ଵ ቁ ቀ1 + ଵ ቁ is 28 ௔ ௔ିଵ ௔ିଶ ௔ିଷ ௔ିସ A) ௔ିଷ B) ௔ାଵ C) ௔ାଵ D) ௔ିଵ ௔ ௔ିସ ௔ିଵ ௔ିସ 14. ቀ1 െ ଵቁ ቀ1 െ ଵ ቁ ቀ1 െ ଵ ቁ………ቀ1 െ ଵ ቁ is C) ௬ିଵ D) ଶ௬ ௬ ௬ାଵ ௬ାଶ ௬ା௬ ଶ௬ ௬ିଵ A) ଵ B) ଵ C) ‫ݐ‬ଷ െ ቀଵቁ + 20 D) None ௬ ଶ௬ ௧ 15. Which of the following expression is a polynomial? A) ‫଺ݔ‬ െ భ + 3 B) 7‫ݕ‬ଽ െ 13‫ݕ‬ଶ + ‫ݕ‬ ‫ݔ‬ల 16. What is the sum of the degree of the polynomial given by ‫ = )ݔ(݌‬2 െ ‫ݔ‬ଶ + ‫ ݔ‬and ‫ = )ݐ(ݍ‬గ ‫ ଼ݐ‬+ ‫ݐ‬ ଶ A) 12 B) 10 C) 11 D) 9 17. If the degree of polynomial ‫ݔܽ = )ݔ(݌‬ଶ െ 2‫ ݔ‬+ 1 is 2 then the value of ‘ܽ’ must be A) rational B) irrational C) real – {0} D) Natural number 18. Which of the following algebraic expression is a polynomial in variable ‫?ݕ‬ A) ‫ݕ‬ଷ + ଵ B) ඥ‫ݕ‬ + ଵ െ 1 ௬య ξ௬ య D) None C) ‫ݕ‬ସ + ‫ݕ‬మ + ‫ݕ‬ 19. If the LCM of the polynomials (‫ ݕ‬െ 3)௔(2‫ ݕ‬+ 1)௕(‫ ݕ‬+ 13)଻ and (‫ ݕ‬െ 3)ସ(2‫ ݕ‬+ 1)ଽ(‫ ݕ‬+ 13)௖ is (‫ ݕ‬െ 3)଺(2‫ ݕ‬+ 1)ଵ଴(‫ ݕ‬+ 13)଻, then the least value of ܽ + ܾ + ܿ is A) 23 B) 3 C) 10 D) 16 20. The polynomial ‫ݔ‬ହ െ ܽଶ‫ݔ‬ଷ െ ‫ݔ‬ଶ‫ݕ‬ଷ + ܽଶ‫ݕ‬ଷ on factorization gives A) (‫ ݔ‬െ ‫ ݔ()ݕ‬െ ܽ)(‫ ݔ‬+ ܽ)(‫ݔ‬ଶ + ‫ݕ‬ଶ + ‫)ݕݔ‬ B) (‫ ݔ‬+ ܽ)(‫ ݔ‬െ ‫ ݔ()ݕ‬െ ܽ)(‫ݔ‬ଶ െ ‫ݕ‬ଶ + ‫)ݕݔ‬ C) (‫ ݔ‬+ ܽ)(‫ ݔ‬+ ‫ ݔ()ݕ‬െ ܽ)(‫ݔ‬ଶ + ‫ݕ‬ଶ + ‫)ݕݔ‬ D) None 21. The LCM of two polynomials ‫ )ݔ(݌‬and ‫ )ݔ(ݍ‬is (‫ ݔ‬+ 3)(‫ ݔ‬െ 2)ଶ(‫ ݔ‬െ 6) and their HCF if (‫ ݔ‬െ 2); If ‫ ݔ( = )ݔ(݌‬+ 3)(‫ ݔ‬െ 2)ଶ, then ‫?= )ݔ(ݍ‬ A) (‫ ݔ‬+ 3)(‫ ݔ‬െ 2) B) ‫ݔ‬ଶ െ 3‫ ݔ‬െ 18 C) ‫ݔ‬ଶ െ 8‫ ݔ‬+ 12 D) ‫ݔ‬ଶ െ 4‫ ݔ‬+ 4 22. If ‫ݔ‬௡ + 1 is divisible by ‫ ݔ‬+ 1, ݊ must be B) an odd natural number D) none A) any natural number C) an even natural number 23. Value of (ଶ.଻)యା(ଵ.ଷ)యା(ଶ)యିଷ×ଵ.ଷ×ଶ.଻×ଶ ଶ.଻×ଵ.ଷାଵ.ଷ×ଶାଶ×ଶ.଻ି(ଶ.଻)మି(ଵ.ଷ)మିଶమ A) 6 B) െ6 C) 10 D) 9 24. Simplify the expression. ቀ௫ + ௬ቁ ቀ௫ െ ௬ቁ ቀ௫మ + ௬మቁ ቀ௫ర + ௬రቁ ቀ ௫ఴ + ௬ఴ ቁ ଶ ଶ ଶ ଶ ସ ସ ଵ଺ ଵ଺ ଶହ଺ ଶହ଺ A) ௫భల െ ௬భల B) ௫ఴ െ ௬ఴ C) ௫భల െ ௬భల D) ௫ఴ െ ௬ఴ ଶହ଺మ ଶହ଺మ ଶହ଺మ ଶହ଺మ ଶହ଺ ଶହ଺ ଶହ଺ ଶହ଺ 25. The polynomial ܽ‫ݔ‬ଷ + 10‫ݔ‬ଶ െ 3 and 3‫ݔ‬ଷ െ 11‫ ݔ‬+ 29 when divided by ‫ ݔ‬െ 2 leaves the remainder ܴ, and ܴଶ respectively then the value of ܽ if ܴଵ െ ܴଶ = 0 A) െ ଷ B) ଷ C) െ ଷ D) None ସ ଻ ଼ 26. If 2‫ݔ‬ଷ + ܽ‫ݔ‬ଶ + 5‫ ݔ‬െ 6 has (‫ ݔ‬െ 1) as a factor and leaves a remainder 2 when divided by (‫ ݔ‬െ 2), find the value of ‘ܽ’ and ‘ܾ’ respectively? A) 8, െ12 B) െ4, 10 C) െ8, 12 D) None Avanti – 9M02 – Polynomials – Olympiad Practice Problems ss


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