# Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not:

Question:

Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not:

$f(x)=x^{5}+3 x^{4}-x^{3}-3 x^{2}+5 x+15, g(x)=x+3$

Solution:

Here, $f(x)=x^{5}+3 x^{4}-x^{3}-3 x^{2}+5 x+15$

g(x) = x + 3

To prove that g(x) is the factor of f(x),

we should show ⟹ f(-3) = 0

here, x + 3 = 0

⟹ x = -3

Substitute the value of x in f(x)

$f(-3)=(-3)^{5}+3(-3)^{4}-(-3)^{3}-3(-3)^{2}+5(-3)+15$

= – 243 + 243 + 27 - 27 - 15 + 15

= 0

Since, the result is 0 g(x) is the factor of f(x)