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A polynomial function $g(x)$ with integer coefficients has a leading coefficient of $-5$ and a constant term of $-2$ . According to the Rational Root Theorem, which of the following are possible roots of $g(x)$ ? $\newline$ Multi-select Choices: $\newline$ (A) $-1$ $\newline$ (B) $-2$ $\newline$ (C) $-\frac{1}{5}$ $\newline$ (D) $1$

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Rational root theorem

Full solution

Q. A polynomial function

g(x)

with integer coefficients has a leading coefficient of

-5

and a constant term of

-2

. According to the Rational Root Theorem, which of the following are possible roots of

g(x)

?

\newline

Multi-select Choices:

\newline

(A)

-1

\newline

(B)

-2

\newline

(C)

-\frac{1}{5}

\newline

(D)

1

Understand Rational Root Theorem: The Rational Root Theorem states that any rational root, in the form of $\frac{p}{q}$ (where $p$ and $q$ are integers), of a polynomial equation with integer coefficients must be such that $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.
Factors of Constant Term: List the factors of the constant term $-2$ : $\pm 1$ , $\pm 2$ .
Factors of Leading Coefficient: List the factors of the leading coefficient $-5$ : $\pm 1$ , $\pm 5$ .
Generate Possible Rational Roots: Generate the possible rational roots by combining the factors of the constant term with the factors of the leading coefficient: $\pm\frac{1}{1}$ , $\pm\frac{2}{1}$ , $\pm\frac{1}{5}$ , $\pm\frac{2}{5}$ .
Simplify Roots: Simplify the possible rational roots: $-1$ , $1$ , $-2$ , $2$ , $-\frac{1}{5}$ , $\frac{1}{5}$ , $-\frac{2}{5}$ , $\frac{2}{5}$ .
Match with Given Choices: Match the simplified possible roots with the given choices: $(A) -1, (B) -2, (C) -\frac{1}{5}, (D) 1$ .

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