Class 10 Maths Problems for Polynomials

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

Answer

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

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

Answer

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

Answer

So it is not a factor

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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.

Answer

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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.

Answer

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

Answer

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

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

Answer

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

Answer

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

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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.

Answer

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

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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.

Answer

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

Now required Quadratic Polynomial
g(x) = x² -(Sum of Zeroes)x +(Product of Zeroes)
=x² – (25/2)x + (157/2)
=2x² – 25x + 157

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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.

Answer

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

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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.

Answer

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

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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 $x^2 - 4x + 3$?

Answer

Using division algorithm

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

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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).

Answer

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

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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.

Answer

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

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

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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.

Also Read

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• Notes
  • Polynomials Class 10 Notes
  • Polynomial Formula Class 10 Cheat sheet
• NCERT Solutions
  • NCERT Solutions Polynomials Class 10 Exercise 2.1
  • NCERT Solutions Polynomials Class 10 Exercise 2.2
  • NCERT Solutions Polynomials Class 10 Exercise 2.3
  • NCERT Solutions Polynomials Class 10 Exercise 2.4
• Assignments
  • Polynomials Class 10 Problem and Solutions
  • Polynomial Class 10 worksheets
  • Polynomial questions for Class 10

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Books Recommended

1. Arihant I-Succeed CBSE Sample Paper Class 10th (2024-2025)
2. Oswaal CBSE Question Bank Class 10 Mathematics (Standard) (2024-2025)
3. PW CBSE Question Bank Class 10 Mathematics with Concept Bank (2024-2025)
4. Bharati Bhawan Secondary School Mathematics CBSE for Class 10th - (2024-25) Examination..By R.S Aggarwal.

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