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A polynomial function $h(x)$ with integer coefficients has a leading coefficient of $7$ and a constant term of $2$ . According to the Rational Root Theorem, which of the following are possible roots of $h(x)$ ? $\newline$ Multi-select Choices: $\newline$ (A) $-\frac{1}{13}$ $\newline$ (B) $\frac{18}{7}$ $\newline$ (C) $-\frac{2}{7}$ $\newline$ (D) $\frac{1}{4}$

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Rational root theorem

Full solution

Q. A polynomial function

h(x)

with integer coefficients has a leading coefficient of

7

and a constant term of

2

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

h(x)

?

\newline

Multi-select Choices:

\newline

(A)

-\frac{1}{13}

\newline

(B)

\frac{18}{7}

\newline

(C)

-\frac{2}{7}

\newline

(D)

\frac{1}{4}

Understand Rational Root Theorem: The Rational Root Theorem states that any rational root, expressed in its simplest form $\frac{p}{q}$ , must have $p$ as a factor of the constant term and $q$ as a factor of the leading coefficient.
Factor Constant Term: List the factors of the constant term $2$ : $\pm1$ , $\pm2$ .
Factor Leading Coefficient: List the factors of the leading coefficient $7$ : $\pm 1$ , $\pm 7$ .
Generate Possible Roots: Generate the possible rational roots by combining the factors of the constant term with the factors of the leading coefficient: $\pm\frac{1}{1}$ , $\pm\frac{1}{7}$ , $\pm\frac{2}{1}$ , $\pm\frac{2}{7}$ .
Simplify Roots List: Simplify the list of possible rational roots: $\pm 1$ , $\pm \frac{1}{7}$ , $\pm 2$ , $\pm \frac{2}{7}$ .
Compare with Given Choices: Compare the simplified list of possible roots with the given choices: $\newline$ (A) $-\frac{1}{13}$ is not a possible root because $13$ is not a factor of the leading coefficient. $\newline$ (B) $\frac{18}{7}$ is not a possible root because $18$ is not a factor of the constant term. $\newline$ (C) $-\frac{2}{7}$ is a possible root because $2$ is a factor of the constant term and $7$ is a factor of the leading coefficient. $\newline$ (D) $\frac{1}{4}$ is not a possible root because $4$ is not a factor of the leading coefficient.

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