# Class 10 Maths Problems for Polynomials

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

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

$p(3) = 3(3)^3-5(3)^2-11 \times 3-3=0$
$p(-1) =3(-1)^3 -5(-1)^2-11(-1)-3=0$
$p(-1/3) =3(-1/3)^3-5(-1/3)^2-11(-1/3)-3=0$

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 $6x^5 + 15x^4 + 16x^3 + 4x^2 + 10x - 35$.

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

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.

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.

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

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.

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

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 α2 + β2 = 24, find the values of k.

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

Now α2 + β2 = 24
(α + β)2 � 2αβ = 24
16/k2 -8/k =24
or
3k2+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.

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(α2 + β2) +13αβ
=6(α + β)2 +αβ=157/2

g(x) = x2 -(Sum of Zeroes)x +(Product of Zeroes)
=x2 – (25/2)x + (157/2)
=2x2 – 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.

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
p2 = 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.

comparing with ax22 + bx + c, we have, a =1 , b= -p & c= -(p+c)
&alpha + β = -b/a = -(-p)/1 = p
αβ = c/a = -(p+c)/1 = -(p+c)
Therefore, (&alpha + 1)*(β+1)
= αβ + α + β + 1
= -(p+c) + p + 1
= -p-c+p+1
= 1-c

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 $x^2 - 4x + 3$?

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).

As per division algorithm
$4x^4 - 5x^3 - 39x^2 -46x- 2= g(x) (x^2 - 3x -5) + (-5x + 8)$
or $g(x) (x^2 - 3x -5) =4x^4 - 5x^3 - 39x^2 - 46x -2 + 5x -8$
$g(x) (x^2 - 3x -5) =4x^4 - 5x^3 - 39x^2 -41x -10$
$g(x) = \frac {4x^4 - 5x^3 - 39x^2 -41x -10}{x^2 - 3x -5}$
Using division method

So $q(x) = 4x^2 + 7x +2$

Question 17
If α and β are the zeroes of the polynomial f(x) = x2 + px + q, form a polynomial whose zeroes are (α + β)2 and (α – β)2.

Given f(x) = x2 + px + q
α + α=-p or (α + β)2=p2
αβ=q
(α – β)2 =(α + β)2 -4αβ= p2 -4q

g(x) = x2 -(Sum of Zeroes)x +(Product of Zeroes)
=x2 -p2x +(p2)(p2 -4q)
=x2 -p2x +p4-4qp2

## Summary

This Class 10 Maths Problems for Polynomials with answers is prepared keeping in mind the latest syllabus of CBSE . This has been designed in a way to improve the academic performance of the students. If you find mistakes , please do provide the feedback on the mail.

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### Practice Question

Question 1 What is $1 - \sqrt {3}$ ?
A) Non terminating repeating
B) Non terminating non repeating
C) Terminating
D) None of the above
Question 2 The volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is?
A) 19.4 cm3
B) 12 cm3
C) 78.6 cm3
D) 58.2 cm3
Question 3 The sum of the first three terms of an AP is 33. If the product of the first and the third term exceeds the second term by 29, the AP is ?
A) 2 ,21,11
B) 1,10,19
C) -1 ,8,17
D) 2 ,11,20