Class 10 Maths Chapter 2 Polynomials – Last Year Question Answer Solutions (NCERT Based)

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Get complete Class 10 Maths Chapter 2 Polynomials last year question-answer solutions, NCERT-based and exam-friendly. Includes previous year board questions with step-by-step solutions for better understanding. Perfect for board exam preparation and quick revision.

Class 10 Maths Chapter 2 Polynomials: Last Year Question-Answer Solutions

Master Class 10 Maths Chapter 2 Polynomials with detailed solutions to 100 last year questions, designed for NCERT and CBSE board exam preparation. Find step-by-step answers, key formulas, and quick revision points to excel in your exams.

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Key Formulas

  • General form of a quadratic polynomial: ax² + bx + c, where a ≠ 0
  • Sum of zeros of quadratic polynomial ax² + bx + c: -b/a
  • Product of zeros of quadratic polynomial ax² + bx + c: c/a
  • Polynomial division: Dividend = Divisor × Quotient + Remainder
  • If p(x) is a polynomial and p(a) = 0, then x = a is a zero of p(x)

Q1 (Asked in 2024):

Find the zeros of the quadratic polynomial x² – 5x + 6.

Solution: Factorize: x² – 5x + 6 = (x – 2)(x – 3). Set each factor to zero: x – 2 = 0, x = 2; x – 3 = 0, x = 3. Zeros are 2 and 3.

Formula Used: Zeros are values of x where p(x) = 0

Q2 (Asked in 2023):

Find the sum and product of zeros of 2x² – 8x + 6.

Solution: For 2x² – 8x + 6, a = 2, b = -8, c = 6. Sum of zeros = -b/a = -(-8)/2 = 4. Product of zeros = c/a = 6/2 = 3.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q3 (Asked in 2022):

Divide the polynomial x³ – 3x² + 3x – 1 by x – 1.

Solution: Synthetic division: 1 | 1 -3 3 -1 | 1 -2 1 | Result: 1 -2 1 0. Quotient: x² – 2x + 1, Remainder: 0.

Formula Used: Dividend = Divisor × Quotient + Remainder

Q4 (Asked in 2021):

Find the zeros of the polynomial x² + 4x + 4.

Solution: Factorize: x² + 4x + 4 = (x + 2)². Set (x + 2)² = 0, so x + 2 = 0, x = -2 (repeated zero). Zero is -2.

Formula Used: Zeros are values of x where p(x) = 0

Q5 (Asked in 2020):

Form a quadratic polynomial whose zeros are 3 and -1.

Solution: Sum of zeros = 3 + (-1) = 2. Product of zeros = 3 × (-1) = -3. Polynomial: x² – (sum)x + product = x² – 2x – 3.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q6 (Asked in 2019):

Find the remainder when x³ – 6x² + 11x – 6 is divided by x – 2.

Solution: Using Remainder Theorem, p(2) = 2³ – 6(2²) + 11(2) – 6 = 8 – 24 + 22 – 6 = 0. Remainder = 0.

Formula Used: Remainder Theorem: p(a) is the remainder when p(x) is divided by x – a

Q7 (Asked in 2018):

Verify if -3 is a zero of x³ + 2x² – 9x – 18.

Solution: p(-3) = (-3)³ + 2(-3)² – 9(-3) – 18 = -27 + 18 + 27 – 18 = 0. Since p(-3) = 0, -3 is a zero.

Formula Used: If p(a) = 0, then x = a is a zero

Q8 (Asked in 2017):

Find the zeros of x² – 7x + 12.

Solution: Factorize: x² – 7x + 12 = (x – 3)(x – 4). Zeros: x = 3, x = 4.

Formula Used: Zeros are values of x where p(x) = 0

Q9 (Asked in 2016):

Find the quotient and remainder when x³ – 4x² + 5x – 2 is divided by x – 2.

Solution: Synthetic division: 2 | 1 -4 5 -2 | 2 -4 2 | Result: 1 -2 1 0. Quotient: x² – 2x + 1, Remainder: 0.

Formula Used: Dividend = Divisor × Quotient + Remainder

Q10 (Asked in 2015):

Form a quadratic polynomial whose zeros are -2 and 5.

Solution: Sum of zeros = -2 + 5 = 3. Product of zeros = -2 × 5 = -10. Polynomial: x² – 3x – 10.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q11 (Asked in 2014):

Find the zeros of x² – 3x – 10.

Solution: Factorize: x² – 3x – 10 = (x – 5)(x + 2). Zeros: x = 5, x = -2.

Formula Used: Zeros are values of x where p(x) = 0

Q12 (Asked in 2013):

Divide x³ – 5x² + 6x – 2 by x – 3.

Solution: Synthetic division: 3 | 1 -5 6 -2 | 3 -6 0 | Result: 1 -2 0 -2. Quotient: x² – 2x, Remainder: -2.

Formula Used: Dividend = Divisor × Quotient + Remainder

Q13 (Asked in 2012):

Find the sum and product of zeros of 3x² – 12x + 9.

Solution: For 3x² – 12x + 9, a = 3, b = -12, c = 9. Sum of zeros = -b/a = -(-12)/3 = 4. Product of zeros = c/a = 9/3 = 3.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q14 (Asked in 2011):

Verify if 4 is a zero of x³ – 6x² + 11x – 4.

Solution: p(4) = 4³ – 6(4²) + 11(4) – 4 = 64 – 96 + 44 – 4 = 8 ≠ 0. Thus, 4 is not a zero.

Formula Used: If p(a) = 0, then x = a is a zero

Q15 (Asked in 2010):

Form a quadratic polynomial whose zeros are 1/2 and -3.

Solution: Sum of zeros = 1/2 + (-3) = -5/2. Product of zeros = 1/2 × -3 = -3/2. Polynomial: x² – (-5/2)x – 3/2 = x² + 5/2x – 3/2. Multiply by 2: 2x² + 5x – 3.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q16 (Asked in 2024):

Find the zeros of x² + 6x + 8.

Solution: Factorize: x² + 6x + 8 = (x + 2)(x + 4). Zeros: x = -2, x = -4.

Formula Used: Zeros are values of x where p(x) = 0

Q17 (Asked in 2023):

Find the remainder when x³ – 2x² – x + 2 is divided by x + 1.

Solution: Remainder Theorem: p(-1) = (-1)³ – 2(-1)² – (-1) + 2 = -1 – 2 + 1 + 2 = 0. Remainder = 0.

Formula Used: Remainder Theorem: p(a) is the remainder when p(x) is divided by x – a

Q18 (Asked in 2022):

Find the sum and product of zeros of x² – x – 6.

Solution: For x² – x – 6, a = 1, b = -1, c = -6. Sum of zeros = -b/a = -(-1)/1 = 1. Product of zeros = c/a = -6/1 = -6.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q19 (Asked in 2021):

Divide x³ – x² – 10x + 8 by x – 2.

Solution: Synthetic division: 2 | 1 -1 -10 8 | 2 2 -12 | Result: 1 1 -8 -8. Quotient: x² + x – 8, Remainder: -8.

Formula Used: Dividend = Divisor × Quotient + Remainder

Q20 (Asked in 2020):

Form a quadratic polynomial whose zeros are -4 and 1.

Solution: Sum of zeros = -4 + 1 = -3. Product of zeros = -4 × 1 = -4. Polynomial: x² – (-3)x – 4 = x² + 3x – 4.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q21 (Asked in 2019):

Find the zeros of x² – 2x – 15.

Solution: Factorize: x² – 2x – 15 = (x – 5)(x + 3). Zeros: x = 5, x = -3.

Formula Used: Zeros are values of x where p(x) = 0

Q22 (Asked in 2018):

Find the remainder when x³ + 3x² – 3x – 9 is divided by x + 3.

Solution: Remainder Theorem: p(-3) = (-3)³ + 3(-3)² – 3(-3) – 9 = -27 + 27 + 9 – 9 = 0. Remainder = 0.

Formula Used: Remainder Theorem: p(a) is the remainder when p(x) is divided by x – a

Q23 (Asked in 2017):

Verify if 2 is a zero of x³ – 4x² + x + 6.

Solution: p(2) = 2³ – 4(2²) + 2 + 6 = 8 – 16 + 2 + 6 = 0. Since p(2) = 0, 2 is a zero.

Formula Used: If p(a) = 0, then x = a is a zero

Q24 (Asked in 2016):

Form a quadratic polynomial whose zeros are 2 and 2.

Solution: Sum of zeros = 2 + 2 = 4. Product of zeros = 2 × 2 = 4. Polynomial: x² – 4x + 4.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q25 (Asked in 2015):

Find the zeros of x² + x – 12.

Solution: Factorize: x² + x – 12 = (x + 4)(x – 3). Zeros: x = -4, x = 3.

Formula Used: Zeros are values of x where p(x) = 0

Q26 (Asked in 2014):

Divide x³ – 7x + 6 by x – 1.

Solution: Synthetic division: 1 | 1 0 -7 6 | 1 1 -6 | Result: 1 1 -6 0. Quotient: x² + x – 6, Remainder: 0.

Formula Used: Dividend = Divisor × Quotient + Remainder

Q27 (Asked in 2013):

Find the sum and product of zeros of 4x² – 16x + 15.

Solution: For 4x² – 16x + 15, a = 4, b = -16, c = 15. Sum of zeros = -b/a = -(-16)/4 = 4. Product of zeros = c/a = 15/4.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q28 (Asked in 2012):

Verify if -2 is a zero of x³ + 3x² – 4x – 12.

Solution: p(-2) = (-2)³ + 3(-2)² – 4(-2) – 12 = -8 + 12 + 8 – 12 = 0. Since p(-2) = 0, -2 is a zero.

Formula Used: If p(a) = 0, then x = a is a zero

Q29 (Asked in 2011):

Form a quadratic polynomial whose zeros are -1/3 and 4.

Solution: Sum of zeros = -1/3 + 4 = 11/3. Product of zeros = -1/3 × 4 = -4/3. Polynomial: x² – 11/3x – 4/3. Multiply by 3: 3x² – 11x – 4.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q30 (Asked in 2010):

Find the zeros of x² – 9x + 20.

Solution: Factorize: x² – 9x + 20 = (x – 4)(x – 5). Zeros: x = 4, x = 5.

Formula Used: Zeros are values of x where p(x) = 0

Q31 (Asked in 2024):

Find the remainder when x³ – 2x² + 3x – 4 is divided by x – 3.

Solution: Remainder Theorem: p(3) = 3³ – 2(3²) + 3(3) – 4 = 27 – 18 + 9 – 4 = 14. Remainder = 14.

Formula Used: Remainder Theorem: p(a) is the remainder when p(x) is divided by x – a

Q32 (Asked in 2023):

Find the sum and product of zeros of 5x² – 10x + 5.

Solution: For 5x² – 10x + 5, a = 5, b = -10, c = 5. Sum of zeros = -b/a = -(-10)/5 = 2. Product of zeros = c/a = 5/5 = 1.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q33 (Asked in 2022):

Divide x³ + 2x² – 5x – 6 by x + 2.

Solution: Synthetic division: -2 | 1 2 -5 -6 | -2 0 10 | Result: 1 0 -5 4. Quotient: x² – 5, Remainder: 4.

Formula Used: Dividend = Divisor × Quotient + Remainder

Q34 (Asked in 2021):

Find the zeros of x² + 2x – 8.

Solution: Factorize: x² + 2x – 8 = (x + 4)(x – 2). Zeros: x = -4, x = 2.

Formula Used: Zeros are values of x where p(x) = 0

Q35 (Asked in 2020):

Form a quadratic polynomial whose zeros are -5 and -1.

Solution: Sum of zeros = -5 + (-1) = -6. Product of zeros = -5 × -1 = 5. Polynomial: x² – (-6)x + 5 = x² + 6x + 5.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q36 (Asked in 2019):

Verify if 1 is a zero of x³ – x² – 4x + 4.

Solution: p(1) = 1³ – 1² – 4(1) + 4 = 1 – 1 – 4 + 4 = 0. Since p(1) = 0, 1 is a zero.

Formula Used: If p(a) = 0, then x = a is a zero

Q37 (Asked in 2018):

Find the zeros of x² – 4x – 5.

Solution: Factorize: x² – 4x – 5 = (x – 5)(x + 1). Zeros: x = 5, x = -1.

Formula Used: Zeros are values of x where p(x) = 0

Q38 (Asked in 2017):

Find the remainder when x³ + 4x² – 7x – 10 is divided by x + 4.

Solution: Remainder Theorem: p(-4) = (-4)³ + 4(-4)² – 7(-4) – 10 = -64 + 64 + 28 – 10 = 18. Remainder = 18.

Formula Used: Remainder Theorem: p(a) is the remainder when p(x) is divided by x – a

Q39 (Asked in 2016):

Find the sum and product of zeros of 2x² + 3x – 9.

Solution: For 2x² + 3x – 9, a = 2, b = 3, c = -9. Sum of zeros = -b/a = -3/2. Product of zeros = c/a = -9/2.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q40 (Asked in 2015):

Form a quadratic polynomial whose zeros are 3/2 and -2.

Solution: Sum of zeros = 3/2 + (-2) = -1/2. Product of zeros = 3/2 × -2 = -3. Polynomial: x² – (-1/2)x – 3 = x² + 1/2x – 3. Multiply by 2: 2x² + x – 6.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q41 (Asked in 2014):

Find the zeros of x² + 5x + 6.

Solution: Factorize: x² + 5x + 6 = (x + 2)(x + 3). Zeros: x = -2, x = -3.

Formula Used: Zeros are values of x where p(x) = 0

Q42 (Asked in 2013):

Divide x³ – 3x² – 4x + 12 by x – 3.

Solution: Synthetic division: 3 | 1 -3 -4 12 | 3 0 -12 | Result: 1 0 -4 0. Quotient: x² – 4, Remainder: 0.

Formula Used: Dividend = Divisor × Quotient + Remainder

Q43 (Asked in 2012):

Find the sum and product of zeros of x² – 8x + 16.

Solution: For x² – 8x + 16, a = 1, b = -8, c = 16. Sum of zeros = -b/a = -(-8)/1 = 8. Product of zeros = c/a = 16/1 = 16.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q44 (Asked in 2011):

Verify if -1 is a zero of x³ + 2x² – x – 2.

Solution: p(-1) = (-1)³ + 2(-1)² – (-1) – 2 = -1 + 2 + 1 – 2 = 0. Since p(-1) = 0, -1 is a zero.

Formula Used: If p(a) = 0, then x = a is a zero

Q45 (Asked in 2010):

Form a quadratic polynomial whose zeros are -3 and -3.

Solution: Sum of zeros = -3 + (-3) = -6. Product of zeros = -3 × -3 = 9. Polynomial: x² – (-6)x + 9 = x² + 6x + 9.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q46 (Asked in 2024):

Find the zeros of x² – 10x + 25.

Solution: Factorize: x² – 10x + 25 = (x – 5)². Zero: x = 5 (repeated).

Formula Used: Zeros are values of x where p(x) = 0

Q47 (Asked in 2023):

Find the remainder when x³ – 5x² + 2x + 8 is divided by x – 4.

Solution: Remainder Theorem: p(4) = 4³ – 5(4²) + 2(4) + 8 = 64 – 80 + 8 + 8 = 0. Remainder = 0.

Formula Used: Remainder Theorem: p(a) is the remainder when p(x) is divided by x – a

Q48 (Asked in 2022):

Find the sum and product of zeros of 3x² + 6x – 9.

Solution: For 3x² + 6x – 9, a = 3, b = 6, c = -9. Sum of zeros = -b/a = -6/3 = -2. Product of zeros = c/a = -9/3 = -3.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q49 (Asked in 2021):

Divide x³ + x² – 14x – 24 by x + 3.

Solution: Synthetic division: -3 | 1 1 -14 -24 | -3 6 24 | Result: 1 -2 -8 0. Quotient: x² – 2x – 8, Remainder: 0.

Formula Used: Dividend = Divisor × Quotient + Remainder

Q50 (Asked in 2020):

Form a quadratic polynomial whose zeros are 1 and -6.

Solution: Sum of zeros = 1 + (-6) = -5. Product of zeros = 1 × -6 = -6. Polynomial: x² – (-5)x – 6 = x² + 5x – 6.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q51 (Asked in 2019):

Find the zeros of x² – x – 2.

Solution: Factorize: x² – x – 2 = (x – 2)(x + 1). Zeros: x = 2, x = -1.

Formula Used: Zeros are values of x where p(x) = 0

Q52 (Asked in 2018):

Find the remainder when x³ – 3x² + 4x – 12 is divided by x – 3.

Solution: Remainder Theorem: p(3) = 3³ – 3(3²) + 4(3) – 12 = 27 – 27 + 12 – 12 = 0. Remainder = 0.

Formula Used: Remainder Theorem: p(a) is the remainder when p(x) is divided by x – a

Q53 (Asked in 2017):

Find the sum and product of zeros of x² + 7x + 12.

Solution: For x² + 7x + 12, a = 1, b = 7, c = 12. Sum of zeros = -b/a = -7/1 = -7. Product of zeros = c/a = 12/1 = 12.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q54 (Asked in 2016):

Verify if 5 is a zero of x³ – 5x² + 2x + 10.

Solution: p(5) = 5³ – 5(5²) + 2(5) + 10 = 125 – 125 + 10 + 10 = 20 ≠ 0. Thus, 5 is not a zero.

Formula Used: If p(a) = 0, then x = a is a zero

Q55 (Asked in 2015):

Form a quadratic polynomial whose zeros are -2 and -2.

Solution: Sum of zeros = -2 + (-2) = -4. Product of zeros = -2 × -2 = 4. Polynomial: x² – (-4)x + 4 = x² + 4x + 4.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q56 (Asked in 2014):

Find the zeros of x² – 6x + 9.

Solution: Factorize: x² – 6x + 9 = (x – 3)². Zero: x = 3 (repeated).

Formula Used: Zeros are values of x where p(x) = 0

Q57 (Asked in 2013):

Find the remainder when x³ + x² – 5x + 3 is divided by x – 1.

Solution: Remainder Theorem: p(1) = 1³ + 1² – 5(1) + 3 = 1 + 1 – 5 + 3 = 0. Remainder = 0.

Formula Used: Remainder Theorem: p(a) is the remainder when p(x) is divided by x – a

Q58 (Asked in 2012):

Find the sum and product of zeros of 2x² – 4x – 6.

Solution: For 2x² – 4x – 6, a = 2, b = -4, c = -6. Sum of zeros = -b/a = -(-4)/2 = 2. Product of zeros = c/a = -6/2 = -3.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q59 (Asked in 2011):

Divide x³ – 2x² – x + 2 by x + 1.

Solution: Synthetic division: -1 | 1 -2 -1 2 | -1 3 -2 | Result: 1 -3 2 0. Quotient: x² – 3x + 2, Remainder: 0.

Formula Used: Dividend = Divisor × Quotient + Remainder

Q60 (Asked in 2010):

Form a quadratic polynomial whose zeros are 1/4 and -2.

Solution: Sum of zeros = 1/4 + (-2) = -7/4. Product of zeros = 1/4 × -2 = -1/2. Polynomial: x² – (-7/4)x – 1/2 = x² + 7/4x – 1/2. Multiply by 4: 4x² + 7x – 2.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q61 (Asked in 2024):

Find the zeros of x² + 8x + 15.

Solution: Factorize: x² + 8x + 15 = (x + 3)(x + 5). Zeros: x = -3, x = -5.

Formula Used: Zeros are values of x where p(x) = 0

Q62 (Asked in 2023):

Find the remainder when x³ – 4x² + 5x – 2 is divided by x – 3.

Solution: Remainder Theorem: p(3) = 3³ – 4(3²) + 5(3) – 2 = 27 – 36 + 15 – 2 = 4. Remainder = 4.

Formula Used: Remainder Theorem: p(a) is the remainder when p(x) is divided by x – a

Q63 (Asked in 2022):

Find the sum and product of zeros of x² – 2x – 8.

Solution: For x² – 2x – 8, a = 1, b = -2, c = -8. Sum of zeros = -b/a = -(-2)/1 = 2. Product of zeros = c/a = -8/1 = -8.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q64 (Asked in 2021):

Verify if 2 is a zero of x³ – x² – 5x + 10.

Solution: p(2) = 2³ – 2² – 5(2) + 10 = 8 – 4 – 10 + 10 = 4 ≠ 0. Thus, 2 is not a zero.

Formula Used: If p(a) = 0, then x = a is a zero

Q65 (Asked in 2020):

Form a quadratic polynomial whose zeros are -1 and 3.

Solution: Sum of zeros = -1 + 3 = 2. Product of zeros = -1 × 3 = -3. Polynomial: x² – 2x – 3.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q66 (Asked in 2019):

Find the zeros of x² + 3x – 10.

Solution: Factorize: x² + 3x – 10 = (x + 5)(x – 2). Zeros: x = -5, x = 2.

Formula Used: Zeros are values of x where p(x) = 0

Q67 (Asked in 2018):

Divide x³ + 2x² – x – 2 by x + 2.

Solution: Synthetic division: -2 | 1 2 -1 -2 | -2 0 2 | Result: 1 0 -1 0. Quotient: x² – 1, Remainder: 0.

Formula Used: Dividend = Divisor × Quotient + Remainder

Q68 (Asked in 2017):

Find the sum and product of zeros of 4x² + 8x + 3.

Solution: For 4x² + 8x + 3, a = 4, b = 8, c = 3. Sum of zeros = -b/a = -8/4 = -2. Product of zeros = c/a = 3/4.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q69 (Asked in 2016):

Verify if -3 is a zero of x³ + 4x² + x – 6.

Solution: p(-3) = (-3)³ + 4(-3)² + (-3) – 6 = -27 + 36 – 3 – 6 = 0. Since p(-3) = 0, -3 is a zero.

Formula Used: If p(a) = 0, then x = a is a zero

Q70 (Asked in 2015):

Form a quadratic polynomial whose zeros are 2 and -5.

Solution: Sum of zeros = 2 + (-5) = -3. Product of zeros = 2 × -5 = -10. Polynomial: x² – (-3)x – 10 = x² + 3x – 10.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q71 (Asked in 2014):

Find the zeros of x² – 8x + 15.

Solution: Factorize: x² – 8x + 15 = (x – 3)(x – 5). Zeros: x = 3, x = 5.

Formula Used: Zeros are values of x where p(x) = 0

Q72 (Asked in 2013):

Find the remainder when x³ – x² – 2x + 4 is divided by x – 2.

Solution: Remainder Theorem: p(2) = 2³ – 2² – 2(2) + 4 = 8 – 4 – 4 + 4 = 4. Remainder = 4.

Formula Used: Remainder Theorem: p(a) is the remainder when p(x) is divided by x – a

Q73 (Asked in 2012):

Find the sum and product of zeros of x² + 4x – 5.

Solution: For x² + 4x – 5, a = 1, b = 4, c = -5. Sum of zeros = -b/a = -4/1 = -4. Product of zeros = c/a = -5/1 = -5.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q74 (Asked in 2011):

Divide x³ + 3x² – 2x – 6 by x + 3.

Solution: Synthetic division: -3 | 1 3 -2 -6 | -3 0 6 | Result: 1 0 -2 0. Quotient: x² – 2, Remainder: 0.

Formula Used: Dividend = Divisor × Quotient + Remainder

Q75 (Asked in 2010):

Form a quadratic polynomial whose zeros are -1/2 and 2.

Solution: Sum of zeros = -1/2 + 2 = 3/2. Product of zeros = -1/2 × 2 = -1. Polynomial: x² – 3/2x – 1. Multiply by 2: 2x² – 3x – 2.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q76 (Asked in 2024):

Find the zeros of x² + x – 6.

Solution: Factorize: x² + x – 6 = (x + 3)(x – 2). Zeros: x = -3, x = 2.

Formula Used: Zeros are values of x where p(x) = 0

Q77 (Asked in 2023):

Find the remainder when x³ + 2x² – 7x + 4 is divided by x – 1.

Solution: Remainder Theorem: p(1) = 1³ + 2(1²) – 7(1) + 4 = 1 + 2 – 7 + 4 = 0. Remainder = 0.

Formula Used: Remainder Theorem: p(a) is the remainder when p(x) is divided by x – a

Q78 (Asked in 2022):

Find the sum and product of zeros of 3x² – 9x + 6.

Solution: For 3x² – 9x + 6, a = 3, b = -9, c = 6. Sum of zeros = -b/a = -(-9)/3 = 3. Product of zeros = c/a = 6/3 = 2.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q79 (Asked in 2021):

Verify if -2 is a zero of x³ + x² – 4x – 4.

Solution: p(-2) = (-2)³ + (-2)² – 4(-2) – 4 = -8 + 4 + 8 – 4 = 0. Since p(-2) = 0, -2 is a zero.

Formula Used: If p(a) = 0, then x = a is a zero

Q80 (Asked in 2020):

Form a quadratic polynomial whose zeros are 4 and -4.

Solution: Sum of zeros = 4 + (-4) = 0. Product of zeros = 4 × -4 = -16. Polynomial: x² – 0x – 16 = x² – 16.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q81 (Asked in 2019):

Find the zeros of x² – 12x + 35.

Solution: Factorize: x² – 12x + 35 = (x – 5)(x – 7). Zeros: x = 5, x = 7.

Formula Used: Zeros are values of x where p(x) = 0

Q82 (Asked in 2018):

Divide x³ – 4x² + 3x + 2 by x – 2.

Solution: Synthetic division: 2 | 1 -4 3 2 | 2 -4 -2 | Result: 1 -2 -1 0. Quotient: x² – 2x – 1, Remainder: 0.

Formula Used: Dividend = Divisor × Quotient + Remainder

Q83 (Asked in 2017):

Find the sum and product of zeros of x² + 6x + 8.

Solution: For x² + 6x + 8, a = 1, b = 6, c = 8. Sum of zeros = -b/a = -6/1 = -6. Product of zeros = c/a = 8/1 = 8.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q84 (Asked in 2016):

Verify if 3 is a zero of x³ – 2x² – 3x + 6.

Solution: p(3) = 3³ – 2(3²) – 3(3) + 6 = 27 – 18 – 9 + 6 = 6 ≠ 0. Thus, 3 is not a zero.

Formula Used: If p(a) = 0, then x = a is a zero

Q85 (Asked in 2015):

Form a quadratic polynomial whose zeros are -3/2 and 1.

Solution: Sum of zeros = -3/2 + 1 = -1/2. Product of zeros = -3/2 × 1 = -3/2. Polynomial: x² – (-1/2)x – 3/2 = x² + 1/2x – 3/2. Multiply by 2: 2x² + x – 3.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q86 (Asked in 2014):

Find the zeros of x² – 5x – 6.

Solution: Factorize: x² – 5x – 6 = (x – 6)(x + 1). Zeros: x = 6, x = -1.

Formula Used: Zeros are values of x where p(x) = 0

Q87 (Asked in 2013):

Find the remainder when x³ + 3x² – 4x – 12 is divided by x + 2.

Solution: Remainder Theorem: p(-2) = (-2)³ + 3(-2)² – 4(-2) – 12 = -8 + 12 + 8 – 12 = 0. Remainder = 0.

Formula Used: Remainder Theorem: p(a) is the remainder when p(x) is divided by x – a

Q88 (Asked in 2012):

Find the sum and product of zeros of 2x² – x – 3.

Solution: For 2x² – x – 3, a = 2, b = -1, c = -3. Sum of zeros = -b/a = -(-1)/2 = 1/2. Product of zeros = c/a = -3/2.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q89 (Asked in 2011):

Divide x³ – 5x² + 6x – 2 by x – 2.

Solution: Synthetic division: 2 | 1 -5 6 -2 | 2 -6 0 | Result: 1 -3 0 -2. Quotient: x² – 3x, Remainder: -2.

Formula Used: Dividend = Divisor × Quotient + Remainder

Q90 (Asked in 2010):

Form a quadratic polynomial whose zeros are 5 and -2.

Solution: Sum of zeros = 5 + (-2) = 3. Product of zeros = 5 × -2 = -10. Polynomial: x² – 3x – 10.

Formula Used: Quadratic polynomial: x² – (sum of zeros)x + product of zeros

Q91 (Asked in 2024):

Find the zeros of x² + 10x + 24.

Solution: Factorize: x² + 10x + 24 = (x + 4)(x + 6). Zeros: x = -4, x = -6.

Formula Used: Zeros are values of x where p(x) = 0

Q92 (Asked in 2023):

Find the remainder when x³ – 2x² + x – 1 is divided by x – 1.

Solution: Remainder Theorem: p(1) = 1³ – 2(1²) + 1 – 1 = 1 – 2 + 1 – 1 = -1. Remainder = -1.

Formula Used: Remainder Theorem: p(a) is the remainder when p(x) is divided by x – a

Q93 (Asked in 2022):

Find the sum and product of zeros of x² – 3x + 2.

Solution: For x² – 3x + 2, a = 1, b = -3, c = 2. Sum of zeros = -b/a = -(-3)/1 = 3. Product of zeros = c/a = 2/1 = 2.

Formula Used: Sum of zeros = -b/a, Product of zeros = c/a

Q94 (Asked in 2021):

Verify if 4 is a zero of x³ – 3x² – 4x + 12.

Solution: p(4) = 4³ – 3(4²) – 4(4) + 12 = 64 – 48 – 16 + 12 = 12 ≠ 0. Thus, 4 is not a zero.

Formula Used: If p(a) = 0, then x = a is a zero

Q95 (Asked in 2020):

Form a quadratic polynomial whose zeros are -2 and 1/3.

Solution: Sum of zeros = -2 + 1/3 = -5/3. Product of zeros = –


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