In each of the following, use factor theorem to find whether polynomial $g(x)$ is a factor of polynomial $f(x)$ or, not.
$f(x) = x^5 + 3x^4 - x^3 - 3x^2 + 5x + 15, g(x) = x + 3$
Given:
$f(x) = x^5 + 3x^4 - x^3 - 3x^2 + 5x + 15, g(x) = x + 3$
To do:
We have to find whether polynomial $g(x)$ is a factor of polynomial $f(x)$ or, not.
Solution:
We know that, if $g(x)$ is a factor of $f(x)$, then the remainder will be zero.
$f(x) = x^5 + 3x^4 - x^3 - 3x^2 + 5x + 15, g(x) = x + 3 = x-(-3)$
So, the remainder will be $f(-3)$.
$f(-3) = (-3)^5+3(-3)^4-(-3)^3-3(-3)^2 +5(-3)+15$
$= -243+3(81) -(-27)-3(9) -15+15$
$=-243+243+27-27$
$=0$
Therefore, $g(x)$ is a factor of polynomial $f(x)$.
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