Nine Standard >> Remainder theorem

Remainder Theorem

Let P(x) be a polynomial in x of degree\(\geq\)1 and g(x)=x-a is a linear polynomial where a is any real number.

When P(x) is divided by x-a, the remainder=P(a).

Proof: Dividend=divisor \(\times\) q(x)+r(x)
           P(x)=(x-a)q(x)+r(x)
Put x=a
P(a)=(a-a).q(a)+r(a)=0+r(a)
        r(a)=0 or some constant

P(a)=0+r(a)=r(a)=remainder

Question:
Find the remainder when \(P(x)=x^{2}-5x+2\) is divided by x-2

Solution:

\(P(x)=x^{2}-5x+2\), q(x)=x-2, a=2

Remainder=P(2)
              =\(2^2-5 \times 2+2\)
              =4-10+2=-4