Question

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

f (x) = x^3 – 5x^2 - 4x + 20
F. (x + 5)
G. (x - 2)
H. (x - 5)
J. (x + 2)

Answer 1

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

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* 2^2 - 4 * 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* (-5)^2 - 4 * (-5) + 20 \\ &= (-125) - 125 + 20 + 20 \\ &= -210\end{aligned}`.

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