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A polynomial function $g(x)$ with integer coefficients has a leading coefficient of $1$ and a constant term of $14$ . According to the Rational Root Theorem, which of the following are possible roots of $g(x)$ ? $\newline$ Multi-select Choices: $\newline$ (A) $\frac{8}{19}$ $\newline$ (B) $\frac{3}{2}$ $\newline$ (C) $-2$ $\newline$ (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

1

and a constant term of

14

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

g(x)

?

\newline

Multi-select Choices:

\newline

(A)

\frac{8}{19}

\newline

(B)

\frac{3}{2}

\newline

(C)

-2

\newline

(D)

-1

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 and $q$ is not zero), of a polynomial equation with integer coefficients must have $p$ as a factor of the constant term and $q$ as a factor of the leading coefficient.
Leading Coefficient: Since the leading coefficient is $1$ , any rational root must have a denominator of $1$ . This means we are looking for integer roots only.
Constant Term Factors: The constant term is $14$ , so the possible integer factors of $14$ are $\pm1$ , $\pm2$ , $\pm7$ , and $\pm14$ .
Check Given Options: Now we check the given options against the possible factors of $14$ .
(A) $\frac{8}{19}$ is not an integer, so it cannot be a root.
Check Given Options: Now we check the given options against the possible factors of $14$ . $\newline$ (A) $\frac{8}{19}$ is not an integer, so it cannot be a root.(B) $\frac{3}{2}$ is not an integer, so it cannot be a root.
Check Given Options: Now we check the given options against the possible factors of $14$ . $\newline$ (A) $\frac{8}{19}$ is not an integer, so it cannot be a root.(B) $\frac{3}{2}$ is not an integer, so it cannot be a root.(C) $-2$ is an integer and a factor of $14$ , so it could be a root.
Check Given Options: Now we check the given options against the possible factors of $14$ .
(A) $\frac{8}{19}$ is not an integer, so it cannot be a root.
(B) $\frac{3}{2}$ is not an integer, so it cannot be a root.
(C) $-2$ is an integer and a factor of $14$ , so it could be a root.
(D) $-1$ is an integer and a factor of $14$ , so it could be a root.

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