Question:
Find the value of $a$, if $(x+2)$ is a factor of $4 x^{4}+2 x^{3}-3 x^{2}+8 x+5 a$
Solution:
Here, $f(x)=4 x^{4}+2 x^{3}-3 x^{2}+8 x+5 a$
By factor theorem
If (x + 2) is the factor of f(x) then, f(-2) = 0
⟹ x + 2 = 0
⟹ x = -2
Substitute the value of x in f(x)
$f(-2)=4(-2)^{4}+2(-2)^{3}-3(-2)^{2}+8(-2)+5 a$
= 4(16) + 2(-8) - 3(4) - 16 + 5a
= 64 - 16 - 12 - 16 + 5a
= 5a + 20
equate f(-2) to zero
f(-2) = 0
⟹ 5a + 20 = 0
⟹ 5a = - 20
⟹ a = -20/5
⟹ a = - 4
When a = - 4, (x + 2) will be factor of f(x)