## REMAINDER THEOREM AND FACTOR THEOREM

Remainder theorem and Factor theorem deal with polynomial division of higher degrees.We have these theorems in algebra particularly in school algebra

Remainder-theorem

If a polynomial f(x) of degree n, which is divided by (x-a), then the remainder is f(a), which is a constant and f(x) can be written as

f(x) = q(x)(x-a)+f(a)

Where q(x) is the quotient polynomial with degree n-1.

Example 1

Find the remainder using Remainder theorem when the polynomial 3x³-2x²+6x-7 is divided by (x-2)

Solution:

By this theorem(remainder)the remainder is f(2). So let us find the value of f(2)

f(2) = 3(2)³-2(2)²+6(2)-7

= 3(8)-2(4)+6(2)-7

= 24-8+12-7

= 21

The remainder is 21.

Now let us check this using synthetic division.

We got the same remainder as 21

Example 2

Find the remainder using the theorem of remainder when the polynomial 7x⁴-x²-3x+9 divided by (x-6).

Solution:

By the Remainder-theorem,the remainder is f(6).

f(6) = 7(6)⁴-(6)²-3(6)+9

= 7(1296)-36-18+9

= 9072-36-18+9

= 9027

The remainder is 9027

We got same answer in both the methods. So we can use either one of the methods to find the remainder while dividing a polynomial by a binomial.

Factor theorem

"The binomial (x-a) is a factor of the polynomial f(x) if and only if f(a)=0"

In other words, ‘If we divide a polynomial f(x) by (x-a) and get the remainder 0, then (x-a) is called as the factor of f(x)’.

Example 1

Check whether (x-2) is a factor of the polynomial f(x)=x⁴-3x³+2x²+8x-16

Solution: f(2) = (2)⁴-3(2)³+2(2)²+8(2)-16

= 16-24+16+8-16

= 0

Here we get f(2)=0. So (x-2) is a factor.

We will verify the same by the synthetic division method.

Example 2

Check whether (x+4) is a factor of f(x)= x²-8x+16

Solution:

f(-4) = (-4)²-8(4)+16

= 16-32 +16

= 32-32

= 0

So (x+4) is a factor of f(x)

These are the examples in the topic Remainder theorem.

### Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life. They are:

It subtracts sadness and adds happiness in our life.

It divides sorrow and multiplies forgiveness and love.

Some people would not be able to accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When our people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”