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Factor Theorem

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☼ Factor Theorem : If f(x) is completely divisible by (x – a), then f(a) = 0.

Thus, (x – a) is a factor of f(x) $\large \Leftrightarrow$ f(a) = 0. ( In this case, remainder = 0 )

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 = 0 ( i.e. remainder (r) = 0 )

$\therefore$ x + 2 is a factor of 5x² + 3x – 14.