# Remainder Theorem with Proof and Examples

Remainder Theorem : Let $$p(x)$$ be any polynomial of degree greater than or equal to one and let a be any real number. If $$p(x)$$ is divided by the linear polynomial $$x-a$$, then the remainder is $$p(a)$$.

Proof : Let $$p(x)$$ be any polynomial with degree greater than or equal to 1 . Suppose that when $$p(x)$$ is divided by $$x-a$$, the quotient is $$q(x)$$ and the remainder is $$r(x)$$, i.e.,
$p(x)=(x-a) q(x)+r(x)$
Since the degree of $$x-a$$ is 1 and the degree of $$r(x)$$ is less than the degree of $$x-a$$, the degree of $$r(x)=0$$. This means that $$r(x)$$ is a constant, say $$r$$.
So, for every value of $$x, r(x)=r$$.
Therefore, $$\quad p(x)=(x-a) q(x)+r$$
In particular, if $$x=a$$, this equation gives us
\begin{aligned} p(a) &=(a-a) q(a)+r \\ &=r, \end{aligned}
which proves the theorem.

Solution: Here, $$\quad p(x)=x^4+x^3-2 x^2+x+1$$, and the zero of $$x-1$$ is 1 .
So, \quad \begin{aligned} p(1) &=(1)^4+(1)^3-2(1)^2+1+1 \\ &=2 \end{aligned}
So, by the Remainder Theorem, 2 is the remainder when $$x^4+x^3-2 x^2+x+1$$ is divided by $$x-1$$

Solution : As you know, $$q(t)$$ will be a multiple of $$2 t+1$$ only, if $$2 t+1$$ divides $$q(t)$$ leaving remainder zero. Now, taking $$2 t+1=0$$, we have $$t=-\frac{1}{2}$$.
Also, $$\quad q\left(-\frac{1}{2}\right)=4\left(-\frac{1}{2}\right)^3+4\left(-\frac{1}{2}\right)^2-\left(-\frac{1}{2}\right)-1=-\frac{1}{2}+1+\frac{1}{2}-1=0$$ So the remainder obtained on dividing $$q(t)$$ by $$2 t+1$$ is 0 .
So, $$2 t+1$$ is a factor of the given polynomial $$q(t)$$, that is $$q(t)$$ is a multiple of $$2 t+1$$

### Noticed a Mistake

Don't worry about it. You only need to copy the URL and then click the submit button below.

$$cos\:2\theta =cos^2\theta -sin^2\theta$$

# Search us like this in Google 🔍

CBSE Class 12 Maths Solution Abstract Classes NCERT Mathematics Solution Abstract Classes IGNOU Maths Assignment Solution IGNOU PGDAST Abstract Classes IGNOU Physics Abstract Classes

Bookmark This Awesome Website

$$cot\:\theta =\frac{cos\:\theta }{sin\:\theta }$$
Insert math as
$${}$$