Use the Factor Theorem to determine whether x - c is a factor of f. If it is, write f in factored form, that is, write f in the
form f(x) = (x - c)(quotient).
-$$f ( x ) = x ^ { 4 } + 10 x ^ { 3 } + 7 x ^ { 2 } + 61 x - 90 ; c = - 10$$
A) Yes; $\mathrm { f } ( \mathrm { x } ) = ( \mathrm { x } + 10 ) \left( \mathrm { x } ^ { 3 } + 7 \mathrm { x } - 9 \right)$
B) Yes; $f ( x ) = ( x - 10 ) \left( x ^ { 3 } + 7 x + 9 \right)$
C) No
D) Yes; $f ( x ) = ( x + 10 ) \left( x ^ { 3 } + x - 9 \right)$

Question

Quizplus · Accepted Answer

To determine if $x - c$ is a factor of $f$, substitute $c$ into $f(x)$. If $f(c) = 0$, then $x - c$ is a factor. Here, $c = -10$, so we check $f(-10)$. Substituting $-10$ into $f(x)$ gives $f(-10) = (-10)^4 + 10(-10)^3 + 7(-10)^2 + 61(-10) - 90 = 10000 - 10000 + 700 - 610 - 90 = 0$. Since $f(-10) = 0$, $x + 10$ is a factor of $f(x)$, and the correct factored form is given in option A.

Use the Factor Theorem to Determine Whether X - C f(x)=x4+10x3+7x2+61x−90;c=−10f ( x ) = x ^ { 4 } + 10 x ^ { 3 } + 7 x ^ { 2 } + 61 x - 90 ; c = - 10f(x)=x4+10x3+7x2+61x−90;c=−10

Use the Factor Theorem to Determine Whether X - C $f ( x ) = x ^ { 4 } + 10 x ^ { 3 } + 7 x ^ { 2 } + 61 x - 90 ; c = - 10$