# CLASS-10REMINDER THEOREM & FACTOR THEOREM - REMINDER THEOREM

Reminder Theorem

When a polynomial f(x) is divided by (x – α), then the reminder is f(α).

Proof.- Let a polynomial f(x) be divided by (x – α)

Then, by division algorithm, we obtain quotient q(x) and a constant reminder ‘r’ such that : -

f(x) = (x – α) . q(x) + r ……………………(i)

on putting x = α in (i), we get f(α) = r

Hence, reminder = f(α)

Deduction 1; When f(x) is divided by (x + α), then reminder is f(- α).

Proof.– We can write, (x + α) = [x – (- α)]

So, when f(x) is divided by [x – (- α)], then reminder = f(- α)

Deduction 2; When f(x) is divided by (ax + b),

- b

then reminder is f(-----)

a

b                -b

Proof; We can write, (ax + b) = a [x + -----] = a [x – (-----)]

a                  a

-b

so, when f(x) is divided by (ax + b), then reminder is f(-----)

a

Example.1) Using reminder theorem, find the reminder when (2x²- 5x + 3) is divided by (x - 2)

Ans.) Let f(x) = 2x²- 5x + 3

By reminder theorem, on dividing f(x) by (x – 1), we get –

Reminder = f(2) = (2 X 2²) – (5 X 2) + 3

= 8 – 10 + 3

= 11 – 10 = 1

Hence the required reminder is    (Ans.)

Example.2) Use reminder theorem to find the reminder when (2x³- 3x² + 7x – 2) is divided by (x – 1)

Ans.) Let, f(x) = 2x³- 3x²+ 7x – 2

By reminder theorem, on dividing f(x) by (x – 1), we get –

Remainder = f(1) = (2 X 1³) – (3 X 1²) + (7 X 1) – 2

= 2 – 3 + 7 – 2

= 9 – 5 = 4

Hence the required reminder is 4.        (Ans.)