Given below are the Class 10 Maths Problems for Polynomials with solutions
(a) cubic polynomials problems
(b) quadratic polynomials Problems
(c) Word Problems

Question 1 Verify that 3, -1, -1/3 are the zeroes of the cubic polynomial $p(x)=3x^3-5x^2-11x-3$ Solution

Question 2 Verify that -5,1/2,3/4 are zeroes of cubic polynomial $4y^3 + 20y + 2y -3$. Also verify the relationship between the zeroes and the coefficients. Question 3 Using division show that $3x^2 + 5$ is a factor of $6x^5 + 15x^4 + 16x^3 + 4x^2 + 10x - 35$. Solution

Question 4 Using division state whether $2y - 5$ is a factor of $4y^4 - 10y^3 - 10y^2 + 30y - 15$. Solution

So it is not a factor

Question 5 Check whether $g(x) = x^2 - 3$ is a factor of $p(x) = 2x^4 + 3x^3 - 2x^2 - 9x -12$ by applying division algorithm. Solution

Question 6 Check whether $p(x) = x^2 +3x +1$ is a factor of $g(x) = 3x^4+ 5x^3 - 7x^2+ 2x + 2$ by using division algorithm. Solution

Question 7 Find remainder when $x^3 - bx^2 + 5- 2b$ is divided by $x - b$. Solution

Given $p(x) =x^3 - bx^2 + 5- 2b$
By remainder theorem
$p(b) = b^3-b^3+5-2b= 5-2b$

Question 8 Check whether polynomial $x - 3$ is a factor of the polynomial $x^3 - 3x^2 - x + 3$. Verify by division algorithm. Solution

Question 9 If $4x^4 + 7x^3 - 4x^2 - 7x + p$ is completely divisible by $x^3 - x$, then find the value of p. Solution

Let $q(x) =4x^4 + 7x^3 - 4x^2 - 7x + p$
$x^3 - x$
$=x(x-1)(x+1)$
So x=0 is a factor of q(x)
$q(0) = 0 + 0 -0 -0 + p =0$
or p=0

Question 10 If α and β are the zeroes of the quadratic polynomial $f(x) = kx^2 + 4x + 4$ such that α^{2} + β^{2} = 24, find the values of k. Solution

for $f(x) = kx^2 + 4x + 4$
α + β=-4/k
αβ=4/k

Now α^{2} + β2 = 24
(α + β)^{2} – 2αβ = 24
16/k^{2} -8/k =24
or
3k^{2}+k-2=0
or k=-1 or 2/3

Question 11 If α and β are the zeroes of the quadratic polynomial $f(x) = 2x^2 - 5x + 7$, find a polynomial whose zeroes are 2α + 3β and 3α + 2β. Solution

for $f(x) = 2x^2 - 5x + 7$
α + β=5/2
αβ=7/2

Now sum of new zeroes
2α + 3β+3α + 2β=5(α + β)=25/2
Product of Zeroes
(2α + 3β)(3α + 2β)=6(α^{2} + β2) +13αβ
=6(α + β)^{2} +αβ=157/2

Now required Quadratic Polynomial
g(x) = x^{2} -(Sum of Zeroes)x +(Product of Zeroes)
=x^{2} – (25/2)x + (157/2)
=2x^{2} – 25x + 157

Question 12 If the squared difference of the zeroes of the quadratic polynomial
$f(x) = x^2 + px + 45$ is equal to 144, find the value of p. Solution

Let α,β are the roots of the quadratic polynomial $f(x) = x^2 + px + 45$ then
&alpha + β = -p and αβ = 45
Given (α - β)^{2} = 144
or (α + β)^{2} – 4αβ = 144
(–p)^{2} – 4 × 45 = 144
p ^{2} – 180 = 144
p^{2} = 144 + 180 = 324
Thus, the value of p is +18 or -18

Question 13 If α and β are the zeroes of the quadratic polynomial $f(x) = x^2 - p (x + 1) - c$, show that (α + 1) (β + 1) = 1 – c. Solution

Question 14 What must be subtracted from the polynomial $f(x) = x^4 + 2x^3 - 13x^2- 12x + 21$
so that the resulting polynomial is exactly divisible by $x^2 - 4x + 3$? Solution

Using division algorithm

$2x-3$ should be subtracted from polynomial $f(x) = x^4 + 2x^3 - 13x^2- 12x + 21$

Question 15 Verify that 1, 2 and -1/2 are zeroes of $2x^3 - 5x^2 + x + 2$. Also verify the relationship between the zeroes and the coefficients. Question 16 On dividing the polynomial $4x^4 - 5x^3 - 39x^2 -46x- 2$ by the polynomial g(x), the quotient is $x^2 - 3x -5$ and the remainder is $-5x + 8$. Find the polynomial g(x). Solution

Question 17 If α and β are the zeroes of the polynomial f(x) = x^{2} + px + q, form a polynomial whose zeroes are (α + β)^{2} and (α – β)^{2}. Solution