Remainder and Factor Theorems

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 x³-7x-6 divided by x-4 gives x²+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 4³-7(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 4x³ – 5x + 1 is divided by
a) x – 2
b) 2x – 1

Solution:

Let f(x) = 4x³– 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 4x² – px + 7 leaves a remainder of –2 when divided by x – 3. Find the value of p.

Solution:

Let f(x) = 4x²– px + 7

By the Remainder Theorem,

f(3) = –2
4(3)²– 3p + 7 = –2
p = 15