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Gyaan Maths
CBSE · Class 6-12
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CBSE Maths · Class 10

Chapter 2: Polynomials

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Date: ______________
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Zeroes of a polynomial and their relationship with coefficients; division algorithm for polynomials.

Questions

  1. Find the sum and product of the zeroes of 2x2−8x+62x^2 - 8x + 6.
  2. Form a quadratic with zeroes 33 and −4-4.
  3. Find the zeroes of x2−2x−8x^2 - 2x - 8.
  4. Verify sum-product for 3x2−x−43x^2 - x - 4.
  5. If α,β\alpha, \beta are zeroes of x2−5x+6x^2 - 5x + 6, find α2+β2\alpha^2 + \beta^2.
  6. Divide x3−3x2+5x−3x^3 - 3x^2 + 5x - 3 by x−1x-1.
  7. If 22 is a zero of x3−3x2+kx−8x^3 - 3x^2 + kx - 8, find kk.
  8. Find zeroes of 6x2+11x+36x^2 + 11x + 3.
  9. Quadratic with sum of zeroes 14\tfrac{1}{4} and product −1-1.
  10. If zeroes of x2−kx+6x^2 - kx + 6 differ by 11, find kk.

Extra Practice (Optional)

  • Zeroes of x2−7x+12x^2 - 7x + 12.
  • Cubic with zeroes 1,2,31, 2, 3.
  • Quadratic with zeroes 2,−2\sqrt 2, -\sqrt 2.
  • If α+β=6,αβ=8\alpha+\beta=6, \alpha\beta=8, find polynomial.
  • Sum of squares of zeroes of x2+4x+4x^2 + 4x + 4.

Answer Key (For Teachers)

  1. Answer: Sum 44, product 33.
    Working: Sum =−b/a=4= -b/a = 4. → Product =c/a=3= c/a = 3.
  2. Answer: x2+x−12x^2 + x - 12.
    Working: Sum =−1= -1, product =−12= -12. → x2−(sum)x+productx^2 - (\text{sum})x + \text{product}.
  3. Answer: 44 and −2-2.
    Working: Split: (x−4)(x+2)(x-4)(x+2).
  4. Answer: Verified.
    Working: Zeroes: (3x−4)(x+1)⇒4/3,−1(3x-4)(x+1) \Rightarrow 4/3, -1. → Sum =1/3=−b/a= 1/3 = -b/a; product =−4/3=c/a= -4/3 = c/a.
  5. Answer: 1313.
    Working: (α+β)2−2αβ=25−12(\alpha+\beta)^2 - 2\alpha\beta = 25 - 12.
  6. Answer: Quotient x2−2x+3x^2 - 2x + 3, remainder 00.
    Working: Long division or synthetic: p(1)=0p(1)=0, quotient x2−2x+3x^2 - 2x + 3.
  7. Answer: k=6k = 6.
    Working: p(2)=8−12+2k−8=0p(2) = 8 - 12 + 2k - 8 = 0. → k=6k = 6.
  8. Answer: −3/2,−1/3-3/2, -1/3.
    Working: Product 1818, sum 1111: split as 9+29 + 2. → 6x2+9x+2x+3=3x(2x+3)+(2x+3)=(2x+3)(3x+1)6x^2 + 9x + 2x + 3 = 3x(2x+3) + (2x+3) = (2x+3)(3x+1).
  9. Answer: 4x2−x−44x^2 - x - 4.
    Working: x2−14x−1x^2 - \tfrac{1}{4}x - 1; multiply by 44: 4x2−x−44x^2 - x - 4.
  10. Answer: k=5k = 5 (or −5-5 if α=−3\alpha = -3).
    Working: Let zeroes α,α+1\alpha, \alpha+1; product =α(α+1)=6⇒α2+α−6=0⇒(α+3)(α−2)=0= \alpha(\alpha+1) = 6\Rightarrow \alpha^2+\alpha-6=0\Rightarrow (\alpha+3)(\alpha-2)=0. → So α=2\alpha = 2 (positive case): zeroes 2,32, 3; k=k = sum =5= 5.
Gyaan Maths · CBSE Class 10 · Chapter 2
Gyaan Maths — CBSE Class 6-12, built for board-toppers.
v1.1 · Sept 2026