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)