**Polynomials**

**Question 1.****Find the remainder when x ^{3} + 3x^{2} + 3x + 1 is divided by**(i) x + 1

(ii) x-

(iii) x

(iv) x+π

(v) 5+2x

**Solution:**

(1) By remainder theorem, the required remainder is equal to p(-1).

Now, p(x)= x^{3}+3x^{2}+3x+1

p(-1) = (-1)^{3} + 3(-1)^{2} + 3(-1) +1 =-1+3-3+1=0

Hence, the required remainder = p(-1) = 0

**Question 2.****Find the remainder when x ^{3} – ax^{2} + 6x – a is divided by x – a.**

**Solution:**

Let p(x) = x^{3}-ax^{2}+6x-a

By remainder theorem, when p(x) is divided by x – a,

Then,remainder = p(a)

p(a)= a^{3}-a . a^{2}+6a-a

=a^{3} – a^{3} +6a-a = 5a

**Question 3.****Check, whether 7 + 3x is a factor of 3x ^{3} + 7x.**

**Solution:**

Let f(x) = 3x^{3} + 7x and g(x) = 7 + 3x

On putting g(x) = 0, we get

7 + 3x = 0 3x = – 7

=> x =

Thus, zero of g(x) is x =.