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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) $\frac{1}{7}$ $\newline$ (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

-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)

\frac{1}{7}

\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.
Factor Constant Term: List the factors of the constant term $-3$ : $\pm 1$ , $\pm 3$ .
Factor Leading Coefficient: List the factors of the leading coefficient $-7$ : $\pm 1$ , $\pm 7$ .
Generate Possible Rational Roots: Generate the possible rational roots by taking all combinations of the factors of the constant term over the factors of the leading coefficient: $\pm\frac{1}{1}$ , $\pm\frac{1}{7}$ , $\pm\frac{3}{1}$ , $\pm\frac{3}{7}$ .
Simplify Roots List: Simplify the list of possible rational roots: $-1$ , $1$ , $-\frac{1}{7}$ , $\frac{1}{7}$ , $-3$ , $3$ , $-\frac{3}{7}$ , $\frac{3}{7}$ .
Match with Given Choices: Match the simplified list of possible rational roots with the given choices: $(A) -1, (B) -\frac{1}{7}, (C) \frac{1}{7}, (D) 1$ .
Identify Correct Choices: Identify the correct choices from the list that are also given in the multi-select options: $(A) -1, (B) -\frac{1}{7}, (C) \frac{1}{7}, (D) 1$ .

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