Question

find the remainder when
$x {}^{3} + 2x {}^{2} + 2x + 1$
is divided by:-
(1) X+1
(2)x​

Answer 1

For finding the remainder, we need to use remainder theorem.

let p(x)=x

3

−2x

2

+x+1

Hence, to find the remainder on dividing by x−1, we need to equate it to 0 and put that value of x in p(x).

⇒x−1=0

⇒x=1

p(1)=1

3

−2×1

2

+1+1

p(1)=1

Answer 2

1) x³+2x²+2x+1÷x+1

By remainder theorem:

Let the root of p(x) = x³+2x²+2x+1 be x+1=0 i.e x= -1

Therefore,

p(-1) = (-1)³+2(-1)²+2(-1)+1

= -1+2-2+1

= 0

Therefore, remainder= 0

2) x³+2x²+2x+1 ÷ x

By remainder theorem:

Let the root of p(x) = x³+2x²+2x+1 be x=0

Therefore,

p(0) = (0)³+2(0)²+2(0)+1

= 1

Therefore, remainder= 1