Question:
For what value of $a$ is $(x-5)$ a factor of $x^{3}-3 x^{2}+a x-10$
Solution:
Here, $f(x)=x^{3}-3 x^{2}+a x-10$
By factor theorem
If (x - 5) is the factor of f(x) then, f(5) = 0
⟹ x - 5 = 0
⟹ x = 5
Substitute the value of x in f(x)
$f(5)=5^{3}-3(5)^{2}+a(5)-10$
= 125 - (3 * 25) + 5a - 10
= 125 - 75 + 5a - 10
= 5a + 40
Equate f(5) to zero
f(5) = 0
⟹ 5a + 40 = 0
⟹ 5a = - 40
⟹ a = − 40/5
= - 8
When a = - 8, (x - 5) will be factor of f(x)