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A polynomial function $g(x)$ with integer coefficients has a leading coefficient of $-14$ and a constant term of $1$ . According to the Rational Root Theorem, which of the following are possible roots of $g(x)$ ? $\newline$ Multi-select Choices: $\newline$ (A) $\frac{15}{2}$ $\newline$ (B) $\frac{15}{7}$ $\newline$ (C) $-\frac{1}{2}$ $\newline$ (D) $\frac{1}{14}$

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Rational root theorem

Full solution

Q. A polynomial function

g(x)

with integer coefficients has a leading coefficient of

-14

and a constant term of

1

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

g(x)

?

\newline

Multi-select Choices:

\newline

(A)

\frac{15}{2}

\newline

(B)

\frac{15}{7}

\newline

(C)

-\frac{1}{2}

\newline

(D)

\frac{1}{14}

Rational Root Theorem: The Rational Root Theorem states that any rational root, in the form of $\frac{p}{q}$ , must have $p$ as a factor of the constant term and $q$ as a factor of the leading coefficient.
Constant Term Factors: List the factors of the constant term $1$ : $\pm1$ .
Leading Coefficient Factors: List the factors of the leading coefficient $-14$ : $\pm1$ , $\pm2$ , $\pm7$ , $\pm14$ .
Generate Possible Roots: Generate all possible rational roots by combining the factors of the constant term with the factors of the leading coefficient: $\pm\frac{1}{1}$ , $\pm\frac{1}{2}$ , $\pm\frac{1}{7}$ , $\pm\frac{1}{14}$ .
Simplify Roots List: Simplify the list of possible rational roots: $\pm 1$ , $\pm \frac{1}{2}$ , $\pm \frac{1}{7}$ , $\pm \frac{1}{14}$ .
Compare with Given Choices: Compare the simplified list of possible rational roots with the given choices to determine which ones are correct: $\newline$ (A) $\frac{15}{2}$ is not a possible root because $15$ is not a factor of $1$ . $\newline$ (B) $\frac{15}{7}$ is not a possible root because $15$ is not a factor of $1$ . $\newline$ (C) $-\frac{1}{2}$ is a possible root because $-1$ is a factor of $1$ and $2$ is a factor of $15$ $0$ . $\newline$ (D) $15$ $1$ is a possible root because $1$ is a factor of $1$ and $15$ $4$ is a factor of $15$ $0$ .

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