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 1 (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.)
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