In each of the following, use factor theorem to find whether polynomial $g(x)$ is a factor of polynomial $f(x)$ or, not.$f(x) = 3x^4 + 17x^3 + 9x^2 - 7x - 10; g(x) = x + 5$
Given:
$f(x) = 3x^4 + 17x^3 + 9x^2 - 7x - 10; g(x) = x + 5$
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.
$f(x) = 3x^4 + 17x^3 + 9x^2 - 7x - 10; g(x) = x + 5 = x - (-5)$
So, the remainder will be $f(-5)$.
$f(-5) = 3(-5)^4+17(-5)^3+9(-5)^2 -7(-5)-10$
$= 3(625) +17(-125)+9(25) +35-10$
$=1875-2125+225+25$
$=2125-2125$
$=0$
Therefore, $g(x)$ is a factor of polynomial $f(x)$.
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