**Notes**
**NCERT Solutions**
**Assignments**
**Revision Notes**
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
p(3) = 3(3)^{3}-5(3)^{2}-11*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.

Solution
**Question 4** Using division state whether 2y – 5 is a factor of 4y

^{4} – 10y

^{3} – 10y

^{2 }+ 30y – 15.

**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

^{2} – 7x

^{2 }+ 2x + 2 by using division algorithm.

**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.

**Question 9** If 4x

^{4} + 7x

^{3} – 4x

^{2 }– 7x + p is completely divisible by x

^{3} – x, then find the value of p.

**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
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

**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?

**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} – 2 by the polynomial g(x), the quotient is x

^{2} – 3x -5 and the remainder is -5x + 8. Find the polynomial g(x).

**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
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}

**Question 18** On dividing the polynomial 4x

^{4} – 5x

^{3} – 39x

^{2} – 2 by the polynomial g(x), the quotient and remainder were x

^{2} – 3x – 5 and -5x + 8 respectively. Find g(x).

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