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A polynomial function $g(x)$ with integer coefficients has a leading coefficient of $-1$ 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) $1$ $\newline$ (C) $-7$ $\newline$ (D) $-3$

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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

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)

1

\newline

(C)

-7

\newline

(D)

-3

Rational Root Theorem: The Rational Root Theorem states that any rational root, in the form of $\frac{p}{q}$ (where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient), must be an integer since the leading coefficient is $-1$ .
List Factors of Constant Term: List the factors of the constant term, which is $3$ . The factors are $\pm1$ and $\pm3$ .
Identify Possible Rational Roots: Since the leading coefficient is $-1$ , the possible rational roots are the factors of the constant term, which are $\pm 1$ and $\pm 3$ .
Check Choices Against Possible Roots: Check the choices given against the possible roots. The possible roots are $-1$ , $1$ , $-3$ , and $3$ .
Match Possible Roots to Choices: Match the possible roots to the choices given: $(A) -1, (B) 1,$ and $(D) -3$ are the correct matches. Choice $(C) -7$ is not a factor of the constant term, so it cannot be a root.

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