Note:

12 ÷ 4 = 3. As there is zero remainder 3 and 4 are factors of 12. 12 = 3 x 4.

13 ÷ 4 = 3 remainder 1 or = 3¼. As there is a non-zero remainder, 3 and 4 are not a factor pair of 13. 13 = (3 x 4) + 1

dividend = divisor × quotient + remainder

The **Remainder Theorem** says that if we divide a polynomial *f*(*x*) by a linear factor *x*–*a*. then, as a result of the long polynomial division, we end up with some polynomial answer *q*(*x*) and some polynomial remainder *r*(*x*).

Hence, when the divisor is linear, the remainder can be found by using the Remainder Theorem.

For example:

That is x^{3}-7x-6 divided by x-4 gives x^{2}+4x+9 remainder 30.

But if we only want to know if the divisor is a factor of the dividend, or what the remainder is, we do not need to do out the long division.

If we sub in 4 for the x in f(x) we get 4^{3}-7(4)-6 = 64-28-6 = 30.

The **Factor Theorem** is the special case of the remainder theorem where the remainder is zero.

It states that

If f(a) = 0, then (x-a) is a factor of f(x).

Conversely, if (x-a) is a factor of f(x), then f(a) = 0.

When f(a) = 0, then “a” is a root.

**Example:**

Find the remainder when 4x^{3} – 5x + 1 is divided by

a) x – 2

b) 2x – 1

**Solution:**

Let f(x) = 4x^{3}– 5x + 1

a) When f(x) is divided by x – 2, remainder,

R = f(2) = 4(2)^{3}– 5(2) + 1 = 23

b) When f(x) is divided by 2x – 1, remainder,

R = f(1/2) = 4 (1/8) -5(1/2) +1 = -1

**Example:**

The expression 4x^{2} – px + 7 leaves a remainder of –2 when divided by x – 3. Find the value of p.

**Solution:**

Let f(x) = 4x^{2}– px + 7

By the Remainder Theorem,

f(3) = –2

4(3)^{2}– 3p + 7 = –2

p = 15