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)=3x3−5x2−11x−3
Question 2 Verify that -5,1/2,3/4 are zeroes of cubic polynomial 4y3+20y+2y−3. Also verify the relationship between the zeroes and the coefficients. Question 3 Using division show that 3x2+5 is a factor of 6x5+15x4+16x3+4x2+10x−35.
Answer
Question 4 Using division state whether 2y−5 is a factor of 4y4−10y3−10y2+30y−15.
Answer
So it is not a factor
Question 5 Check whether g(x)=x2−3 is a factor of p(x)=2x4+3x3−2x2−9x−12 by applying division algorithm.
Answer
Question 6 Check whether p(x)=x2+3x+1 is a factor of g(x)=3x4+5x3−7x2+2x+2 by using division algorithm.
Answer
Question 7 Find remainder when x3−bx2+5−2b is divided by x−b.
Answer
Given p(x)=x3−bx2+5−2b
By remainder theorem p(b)=b3−b3+5−2b=5−2b
Question 8 Check whether polynomial x−3 is a factor of the polynomial x3−3x2−x+3. Verify by division algorithm.
Answer
Question 9 If 4x4+7x3−4x2−7x+p is completely divisible by x3−x, then find the value of p.
Answer
Let q(x)=4x4+7x3−4x2−7x+p x3−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)=kx2+4x+4 such that α2 + β2 = 24, find the values of k.
Answer
for f(x)=kx2+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)=2x2−5x+7, find a polynomial whose zeroes are 2α + 3β and 3α + 2β.
Answer
for f(x)=2x2−5x+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
Question 12 If the squared difference of the zeroes of the quadratic polynomial f(x)=x2+px+45 is equal to 144, find the value of p.
Answer
Let α,β are the roots of the quadratic polynomial f(x)=x2+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)=x2−p(x+1)−c, show that (α + 1) (β + 1) = 1 – c.
Question 14 What must be subtracted from the polynomial f(x)=x4+2x3−13x2−12x+21
so that the resulting polynomial is exactly divisible by x2−4x+3?
Answer
Using division algorithm 2x−3 should be subtracted from polynomial f(x)=x4+2x3−13x2−12x+21
Question 15 Verify that 1, 2 and -1/2 are zeroes of 2x3−5x2+x+2. Also verify the relationship between the zeroes and the coefficients. Question 16 On dividing the polynomial 4x4−5x3−39x2−46x−2 by the polynomial g(x), the quotient is x2−3x−5 and the remainder is −5x+8. Find the polynomial g(x).
Answer
As per division algorithm 4x4−5x3−39x2−46x−2=g(x)(x2−3x−5)+(−5x+8)
or g(x)(x2−3x−5)=4x4−5x3−39x2−46x−2+5x−8 g(x)(x2−3x−5)=4x4−5x3−39x2−41x−10 g(x)=4x4−5x3−39x2−41x−10x2−3x−5
Using division method
So q(x)=4x2+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.
Now required Quadratic Polynomial
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.
