A polynomial in x leaves remainders 8, 4 when divided by (x + 2) and (x - 2) respectively. If the same polynomialis divided by $$x^{2} - 4$$ then the remainder is

A polynomial in x leaves remainders 8, 4 when divided by (x + 2) and (x - 2) respectively. If the same polynomialis divided by $$x^{2} - 4$$ then the remainder is

Let fn. be f(x).

As f(x) leaves remainder, 8 when divided by (x+2)

Therefore, f(-2)= 8

When f(x) is divided by (x−2), remainder = 4

Therefore, f(2)=4

Now, polynomial is divided by $$x^{2} - 4$$ or (x+2)(x-2).

the remainder will be linear. in the form of ax+b.

f(x)=q(x)(x+2)(x−2)+r(x)

When, x=2,

⇒f(2)=0.(x−2)q(2)+r(2)=4⇒f(1)=0.(x−2)q(1)+r(1)=4

r(2) = 2a+b = 4......(i)

When, x=-2

⇒f(-2)=0.(x+2)q(-2)+r(-2)=8⇒f(-2)=0.(x+2)q(-2)+r(2)=8

r(-2) = (-2a)+b = 8......(i)

Solving above two equations, we get remainder r(x) = 6-x Answer