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Remainder Theorem

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☼ Remainder Theorem : If an expression f(x) is divided by (x – a), then the remainder is f(a).

e.g. Is x + 2 a factor of 5x² – 3x – 14 ?

Soln. Here, f (x) = 5x² – 3x – 14 and x + 2 = 0 $\large \Rightarrow$ x = – 2

Now, f (– 2) = 5( – 2)² – 3( – 2) – 14 = 12 $\ne$ 0. ( i.e. r = 12 $\ne$ 0 )

$\therefore$ x + 2 is not a factor of 5x² + 3x – 14.

Some Important Results :

(i) ( xⁿ – aⁿ) is divisible by ( x – a ) for all values of n $\in$ N.

(ii) ( xⁿ – aⁿ) is divisible by ( x + a ) only for even values of n. ( Where, n $\in$ N )

(iii) ( xⁿ + aⁿ) is divisible by ( x + a ) only when n is odd. ( Where, n $\in$ N )

(iv) ( xⁿ + aⁿ) is never divisible by ( x – a ).