Use the factor theorem to determine if the given binomial is a factor of f (x).
-$$f ( x ) = x ^ { 4 } + 8 x ^ { 3 } + 11 x ^ { 2 } - 11 x + 3 ; \quad x + 3$$
A) No
B) Yes

Question

Quizplus · Accepted Answer

The factor theorem states that a polynomial $f(x)$ has a factor $x - c$ if and only if $f(c) = 0$. For the binomial $x + 3$, we set $c = -3$ and substitute it into $f(x)$ to check if the result is 0. Calculating $f(-3) = (-3)^4 + 8(-3)^3 + 11(-3)^2 - 11(-3) + 3 = 81 - 216 + 99 + 33 + 3 = 0$. Since $f(-3) = 0$, $x + 3$ is a factor of $f(x)$.

Use the Factor Theorem to Determine If the Given Binomial f(x)=x4+8x3+11x2−11x+3;x+3f ( x ) = x ^ { 4 } + 8 x ^ { 3 } + 11 x ^ { 2 } - 11 x + 3 ; \quad x + 3f(x)=x4+8x3+11x2−11x+3;x+3

Use the Factor Theorem to Determine If the Given Binomial $f ( x ) = x ^ { 4 } + 8 x ^ { 3 } + 11 x ^ { 2 } - 11 x + 3 ; \quad x + 3$