Remainder theorem of polynomial

In this chapter we will learn to find remainder when one polynomial is divided by another polynomial.

What is remainder theorem of polynomial ?

Let f(x) and g(x) are two polynomials.

Suppose we divide f(x) by g(x).

According to the remainder theorem, if we put root of polynomial g(x) on f(x), the final number we get on calculation will be the remainder of f(x) / g(x).

Let us understand the concept with the help of examples;

Example 01
Consider the two polynomial.
f(x) = \mathtt{\ x^{3} +4x^{2} -3x+10}
g(x) = x + 4

Find the remainder of f(x) / g(x)

Solution
Let’s first find the root of g(x).

x + 4 = 0

x = -4

Hence, -4 is the root of function g(x).

Now put this value in function f(x).

\mathtt{\Longrightarrow \ \ x^{3} +4x^{2} -3x+10}\\\ \\ \mathtt{\Longrightarrow \ ( -4)^{3} +4( -4)^{2} -3( -4) +10}\\\ \\ \mathtt{\Longrightarrow \ 64+64+12+10}\\\ \\ \mathtt{\Longrightarrow \ 22}

Hence, when we divide f(x) by g(x), we will get 22 as remainder.

Question 02
Consider the two functions f(x) and g(x).

f(x) = \mathtt{4x^{4} -3x^{3} -2x^{2} +x-7}

g(x) = x – 1

Find the remainder when f(x) is divided by g(x).

Solution
Let’s first find the root of g(x).

x – 1 = 0

x = 1

Put number 1 in function f(x).

\mathtt{\Longrightarrow \ \ 4x^{4} -3x^{3} -2x^{2} +x-7}\\\ \\ \mathtt{\Longrightarrow \ 4( 1)^{4} -3( 1)^{3} -2( 1)^{2} +1-7}\\\ \\ \mathtt{\Longrightarrow \ 4-3-2+1-7}\\\ \\ \mathtt{\Longrightarrow \ -7}

Hence, we get -7 as remainder.

Question 03
Consider the two function f(x) and g(x).

f(x) = \mathtt{4x^{3} \ –\ 12x^{2} \ +\ 14x\ –\ 3}

g(x) = 2x – 1

Find the remainder when f(x) is divided by g(x).

Solution
Let’s find the root of g(x) first.

2x – 1 = 0

x = 1/2

Put x = 1/2 in function f(x).

\mathtt{\Longrightarrow \ 4x^{3} \ –\ 12x^{2} \ +\ 14x\ –\ 3}\\\ \\ \mathtt{\Longrightarrow \ 4\left(\frac{1}{2}\right)^{3} \ –\ 12\left(\frac{1}{2}\right)^{2} \ +\ 14\left(\frac{1}{2}\right) \ –\ 3}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{8} -\frac{12}{4} +7-3}\\\ \\ \mathtt{\Longrightarrow \frac{1}{2} -3+7-3}\\\ \\ \mathtt{\Longrightarrow \frac{3}{2}}

Hence, 3/2 will be the remainder.

Next chapter : Factor theorem of polynomial