Question

In each of the following, two polynomials, find the value of a, if x + a is a factor.

(i) $x^3+a{x}^2$ - 2x + a + 4

(ii) $x^4-a^2{x}^2$ + 3x - a

Answer 1

(i) Let f(x) = $x^3+a{x}^2$ - 2x + a + 4

and g(x) = x + a

$\therefore$ x + a is a factor of f(x)

Let x + a = 0, then x = - a

$\therefore$ f(-a) $=(-a{)}^3+a(-a{)}^2-2\times$ (-a) + a + 4

= - $a^3+a^3$ + 2a + a + 4

= 3a + 4

$\therefore$ Remainder = 0

$\therefore$ 3a + 4 = 0 $\Rightarrow$ 3a = - 4

$\Rightarrow a=\frac{-4}{3}$

Hence a = $\frac{-4}{3}$

(ii) Let f(x) = $x^4-a^2{x}^2$ + 3x - a

and g(x) = x + a

Let x + a = 0 $\Rightarrow$ x = - a

Now f(-a) $=(-a{)}^4-a^2(-a{)}^2$ + 3(-a) - a

= $a^4-a^4$ - 3a - a

= - 4a

$\therefore$ Remainder = 0

$\therefore$ -4a = 0 $\Rightarrow$ a = 0

Hence a = 0