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A polynomial function $g(x)$ with integer coefficients has a leading coefficient of $-7$ and a constant term of $-3$ . 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) $\frac{1}{7}$ $\newline$ (C) $-3$ $\newline$ (D) $\frac{1}{2}$

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Rational root theorem

Full solution

Q. A polynomial function

g(x)

with integer coefficients has a leading coefficient of

-7

and a constant term of

-3

. 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)

\frac{1}{7}

\newline

(C)

-3

\newline

(D)

\frac{1}{2}

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.
Constant Term Factors: List the factors of the constant term $-3$ : $\pm 1$ , $\pm 3$ .
Leading Coefficient Factors: 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: $\pm\frac{1}{1}$ , $\pm\frac{1}{7}$ , $\pm\frac{3}{1}$ , $\pm\frac{3}{7}$ .
Simplify Roots: Simplify the possible rational roots: $\pm 1$ , $\pm \frac{1}{7}$ , $\pm 3$ , $\pm \frac{3}{7}$ .
Match with Choices: Match the simplified possible roots with the given choices: $(A) \, 1, (B) \, \frac{1}{7}, (C) \, -3, (D) \, \frac{1}{2}$ .
Choice (A): Choice (A) $1$ is a possible root because $1$ is a factor of $-3$ .
Choice (B): Choice (B) $\frac{1}{7}$ is a possible root because $1$ is a factor of $-3$ and $7$ is a factor of $-7$ .
Choice (C): Choice (C) $-3$ is a possible root because $-3$ is a factor of $-3$ .
Choice (D): Choice (D) $\frac{1}{2}$ is not a possible root because $2$ is not a factor of $-7$ .

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