If (x − 1) is a factor of $x^{2} + 4 x + a$ and
(x − 2) is a factor of $x^{2} + 4 x + b$ , then find the value of a + b .
-17

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Solution

The correct option is A -17
According to the factor theorem, if $x - a$ is a factor of a polynomial f( $x$ ), then
f(a) = 0.
Since ( $x - 1$ ) is a factor of $x^{2} + 4 x + a$ ,
$\Rightarrow 1^{2} + 4 \times 1 + a = 0$
$\Rightarrow 1 + 4 + a = 0$
$\Rightarrow a = - 5$

Since ( $x - 2$ ) is a factor of $x^{2} + 4 x + b$ ,
$\Rightarrow$ $2^{2} + 4 \times 2 + b = 0$
$\Rightarrow 4 + 8 + b = 0$
$\Rightarrow b = - 12$

Hence, a + b = (-5) + (-12) = -17.

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