Remainder Theorem

Recall the Remainder theorem and the Factor theorem.

When f(x) is divided by (x-a) the remainder is (x-a) and if the remainder f(a) = 0 then x - a is a factor of the expression f (x).

Remainder theorem

If f(x) is a polynomial in x and is divided by x-a; the remainder is the value of f(x) at x = a i.e., Remainder = f(a).

Proof:

Let p(x) be a polynomial divided by (x-a).

Let q(x) be the quotient and R be the remainder.

By division algorithm,

Dividend = (Divisor x quotient) + Remainder

p(x) = q(x) . (x-a) + R

Substitute x = a,

p(a) = q(a) (a-a) + R

p(a) = R (a - a = 0, 0 - q (a) = 0)

Hence Remainder = p(a).

Example 1:

Find the remainder when 2x³+4 is divided by

a) x-1

b) x-2

d) x+1

Suggested answer:

Let f(x) = 2x³ + 4

a) When f (x) is divided by x-1,

R = f (1)

= 2(1)³+ 4

= 6

b) When f (x) is divided by x-2,

R = f (2)

= 2(2)³+ 4

= 20

d) When f(x) is divided by x+1, i.e., x-(-1).

R = f(-1)

= 2(-1)³ + 4

= -2+4 = 2

Example 2:

Factorise x² - 7x + 10.

Suggested answer:

Let f(x) = x² - 7x + 10

Then f (2) = (2)² - 7(2) +10 [By trial and error method]

To find the other factor of x² - 7x + 10, divide x² - 7x + 10 by (x-2)

Verification:

Example 3:

Factorise x² + 3x -4 using Remainder theorem.

Suggested answer:

Let f(x) = x² + 3x - 4

Step 1:

Find a value of the variable x for which the value of x² + 3x - 4

is equal to 0.

Let x =1,

Then, f(1) = (1)² + 3(1) - 4 = 0

Step 2:

Divide x² + 3x -4 by (x-1),

Step 3:

Example 4:

Factorise 2x³ + x² - 2x - 1.

Suggested answer:

Let f (x) = 2x³ + x² - 2x - 1

= 0

Now, divide 2x³ + x² - 2x - 1 by (x-1),

2x² + 3x + 1 can be further factorised by splitting the middle term.

The factors of 2x³ + x² -2x -1 are (x -1) , (x+1) and (2x+1).

Example 5:

Factorise completely 2x³ - 7x² - 3x + 18.

Suggested answer:

Let f (x) = 2x³ - 7x² - 3x + 18

Let x = 1,

This can be further factorised by splitting the middle term.

Steps for factorisation using remainder theorem

• By trial and error method, find the factor of the constant for which the given expression becomes equal to zero.

• Divide the expression by the factor that is determined in step 1.

• Factorise the quotient. If the quotient is a trinomial, factorise it further.

• If the expression is a 4^th degree expression, the first step will be to reduce this to a trinomial and then factorise this trinomial further.