Question

If a polynomial f(x)​ has a remainder of 3​ when divided by x−4​, what is f(4)​?

Answer 1

Dividend=Divisor×Quotient+Remainder

So, Applying it:−

Let q(x),k(x) be quotient when f(x) is divided by x−1 and x−2 respectively

⇒f(x)=(x−1)q(x)+5

∴f(1)=5 ..... (1)

Also,f(x)=(x−2)k(x)+7

∴f(2)=7 ..... (2)

Now, let ax+b be the remainder when f(x) is divided by (x−1)(x−2) and g(x) be the quotient.

f(x)=(x−1)(x−2)g(x)+(ax+b)

Using (1) and (2)

5=a+b ...... (3)

7=2a+b ...... (4)

Solving (3) and (4), we get

a=2 and b=3

∴2x+3 is the remainder when f(x) is divided by (x−1)(x−2).