Question:
Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not:
$f(x)=3 x^{4}+17 x^{3}+9 x^{2}-7 x-10, g(x)=x+5$
Solution:
Here, $f(x)=3 x^{4}+17 x^{3}+9 x^{2}-7 x-10$
g(x) = x + 5
To prove that g(x) is the factor of f(x),
we should show ⟹ f(-5) = 0
here, x + 5 = 0
⟹ x = - 5
Substitute the value of x in f(x)
$f(-5)=3(-5)^{4}+17(-5)^{3}+9(-5)^{2}-7(-5)-10$
= (3 * 625) + (12 * (-125)) + (9*25) + 35 - 10
= 1875 - 2125 + 225 + 35 - 10
= 2135 - 2135
= 0
Since, the result is 0 g(x) is the factor of f(x)