Question

Which of the following is NOT a linear factor of the polynomial function?

Answer 1

Answer:

Among the four choices, $(x + 5)$ is the only one that is not a linear factor of this polynomial function.

Step-by-step explanation:

Let $a$ denote some constant. A linear factor of the form $(x - a)$ is a factor of a polynomial $f(x)$ if and only if $f(a) = 0$ (that is: replacing all $x$ in the polynomial $f(x) \!$ with the constant $a\!$ would give this polynomial a value of $0$.)

For example, in the second linear factor $(x - 2)$, the value of the constant is $a = 2$. Verify that the value of $f(2)$ is indeed $0$. (In other words, replacing all $x$ in the polynomial $f(x) \!$ with the constant $2$ should give this polynomial a value of $0\!$.)

$\begin{aligned}f(2) &= 2^3 - 5\times 2^2 - 4 \times 2 + 20 \\ &= 8 - 20 - 8 + 20 \\ &= 0 \end{aligned}$.

Hence, $(x - 2)$ is indeed a linear factor of polynomial $f(x)$.

Similarly, it could be verified that $(x - 5)$ and $(x + 2)$ are also linear factors of this polynomial function.

Rewrite the first linear factor $(x + 5)$ in the form $(x - a)$ for some constant $a$: $(x + 5) = (x - (-5))$, where $a = -5$.

Calculate the value of $f(5)$.

$\begin{aligned}f(5) &= (-5)^3 - 5\times (-5)^2 - 4 \times (-5) + 20 \\ &= (-125) - 125 + 20 + 20 \\ &= -210\end{aligned}$.

$f(5) \ne 0$ implies that $(x - (-5))$ (which is equivalent to $(x + 5)$) isn't a linear factor of this polynomial function.