If $(x - a)$ and $(x - 2)$ are the factors of $p (x) = x^{2} - b x + 2$ , what is the value of $a + b$ ?

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Solution

The correct option is D 4
Given ,
$x - 2$ is a factor of $p (x)$
$\Rightarrow$ $p (2)$ = 0
$\Rightarrow$ $4 - 2 b + 2 = 0$
$\Rightarrow$ $b = 3$

$∴$ $p (x) = x^{2} - 3 x + 2$

Splitting the middle term , we have
$x^{2} - 3 x + 2 = x^{2} - 2 x - x + 2$
$\Rightarrow$ $x (x - 2) - 1 (x - 2)$
$\Rightarrow$ $x^{2} - 3 x + 2 = (x - 2) (x - 1)$

Given, $x - 2$ is already a factor
$\Rightarrow$ $x - a$ is equal to $x - 1$
$\Rightarrow$ $a = 1$

$∴$ $a + b = 1 + 3 = 4$

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Find the value of ‘k’ if x -1 is a factor of P(x) = 4 $x^{3}$ + $x^{2}$ – 3x + k.

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