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

Solution:

1. $x^{3}+a x^{2}-2 x+a+4$

let, $f(x)=x^{3}+a x^{2}-2 x+a+4$

here, x + a = 0

⟹ x = - a

Substitute the value of x in f(x)

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

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

= 3a + 4

Equate to zero

⟹ 3a + 4 = 0

⟹ 3a = -4

⟹ a = − 4/3

So, when (x + a) is a factor of f(x) then a = −4/3

2. $x^{4}-a^{2} x^{2}+3 x-a$

let, $f(x)=x^{4}-a^{2} x^{2}+3 x-a$

here, x + a = 0

⟹ x = - a

Substitute the value of x in f(x)

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

$=a^{4}-a^{4}-3(a)-a$

= -4a

Equate to zero

⟹ -4a = 0

⟹ a = 0

So, when (x + a) is a factor of f(x) then a = 0