Worksheet- 3
Polynomials
Name: Grade: X _
Subject: Mathematics Date:
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1. If (x + a) is the factor of the polynomial x2 + px + q and x2 + mx + n, prove that a = .
2. If are the zeros of x2 + 5x + 5, find the value of
3. If are the zeros of the polynomial f(x) = 3x2 + 2x + p and = , then find the value of p.
4. If are the zeros of the polynomial f(x) = 2x2 + 3x - 5, then find the value of ( )2
5. If are the zeros of the polynomial f(x) = x2 + px + q, then find the polynomial having and as its zeros.
6. If are the zeros of the polynomial x2 + 5x +4, then find the value of + -2 .
7. If x2 – 1 is a factor of ax4 + bx3 + cx2 + dx + e, show that a + c +e = b + d =0.
8. If (x - a) is the factor of the polynomial x3 – mx2 – 2nax + na2, prove that a = m + n where a ≠ 0.
9. On dividing 3x3 – 2x2 +5 x + 5 by a polynomial p(x), the quotient and remainder are x2 -x + 2 and 3 respectively.
Find p (x).
10. If the polynomial x4 + 2x3 + 8x2 + 12x + 18 is divided by another polynomial x2 + 5, the remainder comes out to
be (ax + b), find a and b.
11. If are the zeroes of cubic polynomial kx3 - 5x + 9. If 27, find the value of k.
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12. If are the zeros of the cubic polynomial x3 + 4x + 2, then find the value of .
13. If are the zeros of the polynomial f(x) = x2 – px +q, prove that + = - +2.
14. If are the zeros of the polynomial f(x) = x2 - p(x+1) – c, show that (
15. If the squared difference of the zeros of the quadratic polynomial f(x) = x2 + px + 45 is equal to 144, find the
value of p.
16. If the sum of squares of the zeros of the quadratic polynomial f(x) = x2 - 8x + k is 34, then find the value of k.
17. If are zeros of the polynomial x2 – 6x + a. Find the value of a, if 3 = 20.
18. If are the roots of the polynomial p(x) = x2 - (k + 6)x +2(2k - 1). Find the value of k, if
1
19. If one zero of the polynomial ax2 + bx + c is double the other, prove that 2b2 = 9ac.
20. What must be subtracted from 8x4 + 14x3 +x2 + 7x +8 so that polynomial is exactly divisible by 4x2 – 3x +2?
21. What must be added to 4x4 + 2x3 -2x2 + x -1 so that polynomial is exactly divisible by x2 + 2x -3?
22. If are zeros of cubic polynomial x3 + px2 + qx + 2 such that + 1 = 0. Find the value of 2p + q + 5.
23. If and are the zeros of the polynomial kx2 + 4x + 4 and ( )2 - 2 = 24, then find the value of k.
24. If one zero of the polynomial p(x) = (a2 +9)x2 + 45x + 6a is reciprocal of the other, find the value of a.
25. Find the zeros of the polynomial f(x) = x3 – 5x2 -1x + 80, if its two zeros are equal in magnitude but opposite in
sign.
26. If one zero of the polynomial (K + 1)x2 – 5x + 5 is multiplicative inverse of the other, then find the zeroes of kx2
– 3kx + 9, where k is constant.
27. If sum of the zeroes of the polynomial 5x2 – (3 + k) x + 7 is zero, then find the zeros of the polynomial 2x2 – 2(K +
11)x + 30.
28. If the product of the zeroes of the polynomial kx2 + 41x + 42 is 7 then find the zeroes of the polynomial (k – 4)x2 + (k
+ 1)x + 5.
29. If are the zeroes of cubic polynomial x3 – 12x2 + 44x + c if = , find the value of c.
30. If are zeroes of cubic polynomial x3 – 2x2 + qx – r. If = 0 then show that 2q = r.
31. Zeroes of a quadratic polynomial are 3 and 7. Find the remainder when this polynomial is divided by x2 – 5x + 6.
32. P(x) is a polynomial of degree more than 2. When p(x) is divided by x – 2, it leaves remainder 1 and when it is
divided by x – 3 it leaves a remainder 3. Find the remainder when p(x) is divided by (x – 2)(x – 3).
33. If x3 + x2 – ax + b is divided by x2 – x, write the values of ‘a’ and’ b’.
34. If the zeroes of x2 – px + 6 are in the ratio 2 : 3, find p.
35. If are the zeroes of polynomial p(x) = x2 – k(x + 1) – p such that ( + 1)( + 1) = 0, find p.
36. a, b, c are co-prime a ≠ 1 such that 2b = a + c. If ax2 – 2bx + c and 2x3 – 5x2 + kx + 4 has one integral root
common, then find the value of k.
37. If m, n are zeroes of ax2 – 5x + c. Find the value of ‘a’ and ‘c’ if m + n = m.n = 10.
38. If be zeroes of polynomial 6x3 + 3x2 – 5x + 1, then find the value of If -1 + -1 + -1 .
39. If the remainder on division of x3 – kx2 + 13x – 21 by 2x – 1 is -21, find the quotient and the value of ‘k’. Hence,
find the zeroes of the cubic polynomial x3 – kx2 + 13x.
40. Find the zeroes of the polynomial p(x) = x3 – 5x2 – 2x + 24, if it is given that the product of its zeroes is 12.
41. Find the zeroes of the polynomial 2x3 + 5x2 – 9x – 18, if it is given that the product of its two zeroes is -3.
42. If one of the zeroes of the cubic polynomial x3 + ax2 + bx + c is -1, then prove that the product of the two zeroes
is b – a +1.
