Given below are the Class 10 Maths Problems for Polynomials with solutions

(a) cubic polynomials problems

(b) quadratic polynomials Problems

(c) Word Problems

(a) cubic polynomials problems

(b) quadratic polynomials Problems

(c) Word Problems

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

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

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

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

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.

Using division show that $3x^2 + 5$ is a factor of $6x^5 + 15x^4 + 16x^3 + 4x^2 + 10x - 35$.

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

So it is not a factor

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.

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.

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

$p(b) = b^3-b^3+5-2b= 5-2b$

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

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

If α and β are the zeroes of the quadratic polynomial $f(x) = kx^2 + 4x + 4$ such that α

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

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(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

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

p^{2} = 144 + 180 = 324

Thus, the value of p is +18 or -18

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 ax^{2}2 + 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

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 $f(x) = x^4 + 2x^3 - 13x^2- 12x + 21$

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.

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

If α and β are the zeroes of the polynomial f(x) = x

Given f(x) = x^{2} + px + q

α + α=-p or (α + β)^{2}=p^{2}

αβ=q

(α – β)^{2} =(α + β)^{2} -4αβ= p^{2} -4q

Now required Quadratic Polynomial

g(x) = x^{2} -(Sum of Zeroes)x +(Product of Zeroes)

=x^{2} -p^{2}x +(p^{2})(p^{2} -4q)

=x^{2} -p^{2}x +p^{4}-4qp^{2}

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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Class 10 Maths Class 10 Science