# Class 10 NCERT Solutions- Chapter 2 Polynomials – Exercise 2.3

**Question 1. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:**

**(i) p(x) = x ^{3} – 3x^{2} + 5x – 3, g(x) = x^{2} – 2**

**(ii) p(x) = x ^{4} – 3x^{2} + 4x + 5, g(x) = x^{2} + 1 – x**

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**(iii) p(x) = x ^{4}– 5x + 6, g(x) = 2 – x^{2}**

**Solution:**

i)p(x) = x^{3}– 3x^{2}+ 5x – 3, g(x) = x^{2}– 2

R = 7x-9

Q = x-3

ii) p(x) = x^{4}– 3x^{2}+ 4x + 5, g(x) = x^{2}+ 1 – x

R = 8

Q = x^{2}+x-3

iii) p(x) = x^{4}– 5x + 6, g(x) = 2 – x^{2}

Q = -x^{2}-2

R = -5x+10

**Question 2. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.**

**(i) t ^{2} – 3, 2t^{4} + 3t^{3} – 2t^{2}– 9t – 12**

**(ii) x ^{2} + 3x + 1, 3x^{4} + 5x^{3} – 7x^{2} + 2x + 2**

**(iii) x ^{3 }– 3x + 1, x^{5} – 4x^{3} + x^{2} + 3x + 1**

**Solution:**

i) t^{2}– 3, 2t^{4}+ 3t^{3 }– 2t^{2}– 9t – 12

Q = 2t^{3}+3t+4

R = 0Yes 1

^{st}polynomial is factor of 2^{nd}polynomial.

ii) x^{2}+ 3x + 1, 3x^{4}+ 5x^{3}– 7x^{2}+ 2x + 2

R = 0

Q = 3x^{2}-4x+2

iii) x^{3}– 3x + 1, x^{5}– 4x^{3}+ x^{2}+ 3x + 1

R = x^{2}-1

Q = 2

**Question 3. Obtain all other zeroes of 3x**^{4} + 6x^{3} – 2x^{2} – 10x – 5, if two of its zeroes are and √(5/3) and -√(5/3).

^{4}+ 6x

^{3}– 2x

^{2}– 10x – 5, if two of its zeroes are and √(5/3) and -√(5/3).

**Solution:**

R = 0

Q = 3x^{2}+6x+3∴ we are factorizing

3x

^{2}+6x+3x

^{2}+2x+1(x+1)

^{2}(x+1) (x+1) = 0

∴ x = -1 and x = -1

**Question 4. On dividing x**^{3} – 3x^{2} + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and -2x + 4 respectively. Find g(x).

^{3}– 3x

^{2}+ x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and -2x + 4 respectively. Find g(x).

**Solution:**

Dividend = Divisor * Quotient + Remainder

x

^{3}-3x^{2}+3x-2/x-2

R = 0

Q = x^{2 }-x +1

Answer:g(x)=x^{2}-x+1

**Question 5. Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and:**

**(i) deg p(x) = deg q(x)**

**(ii) deg q(x) = deg r(x)**

**(iii) deg r(x) = 0**

**Solution:**

i)deg p(x) = deg q(x)p(x)=2x

^{2}-2x+14, g(x)=2p(x)/g(x)=2x

^{2}-2x+14/2=(x^{2}-x+7)=x

^{2}-x+7=q(x)=q(x)=x

^{2}-x+7

r(x)=0

ii) deg q(x)=deg r(x)p(x)=4x

^{2}+4x+4, g(x)=x^{2}+x+1

q(x) = 4

r(x) = 0∴Here deg q(x)=deg r(x)

iii) deg r(x)=0p(x)=x

^{3}+2x^{2}-x+2 ,g(x)=x^{2}-1

q(x) = x+2

r(x) = 4

deg of r(x) = 0