Question

find the remainder when p(x )is divided by (x+1)​

Answer 1

Answer:

Remainder theorem says that:

“When a polynomial p(x) is divided by (x−a) , the remainder is p(a) .”

Also, when a polynomial p(x) is divided by another polynomial q(x) ,the degree of the remainder is at most 1 less than the degree of q(x) .

Using the remainder theorem, we can write:

p(1)=3;p(3)=5

p(x) can be written as:

Dividend=(Divisor×Quotient)+Remainder

p(x)=(x−1)(x−3)Q(x)+r(x)

r(x) is the remainder polynomial and Q(x) is the Quotient polynomial. Since r(x) is linear, r(x)=Ax+B , where A and B are arbitrary constants.

⟹p(x)=(x−1)(x−3)Q(x)+Ax+B

Now,

p(1)=A+B=3⟹A+B=3

p(3)=3A+B=5⟹3A+B=5

Solving the 2 equations, we get A=1 and B=2 .

Therefore, r(x)=Ax+B=x+2

r(−2)=−2+2=0

Hence, r(−2)=0