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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)

User Rabster
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8.2k points

1 Answer

2 votes

Answer:

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.

User Bhb
by
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