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A polynomial function $h(x)$ with integer coefficients has a leading coefficient of $2$ and a constant term of $-7$ . According to the Rational Root Theorem, which of the following are possible roots of $h(x)$ ? $\newline$ Multi-select Choices: $\newline$ (A) $-1$ $\newline$ (B) $-\frac{1}{2}$ $\newline$ (C) $-7$ $\newline$ (D) $\frac{7}{3}$

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Rational root theorem

Full solution

Q. A polynomial function

h(x)

with integer coefficients has a leading coefficient of

2

and a constant term of

-7

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

h(x)

?

\newline

Multi-select Choices:

\newline

(A)

-1

\newline

(B)

-\frac{1}{2}

\newline

(C)

-7

\newline

(D)

\frac{7}{3}

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 $q \neq 0$ ), of a polynomial equation with integer coefficients is 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 $-7$ : $\pm 1$ , $\pm 7$ .
Leading Coefficient Factors: List the factors of the leading coefficient $2$ : $\pm1$ , $\pm2$ .
Generate Possible Roots: Generate the possible rational roots by taking all combinations of the factors of the constant term over the factors of the leading coefficient: $\pm\frac{1}{1}$ , $\pm\frac{7}{1}$ , $\pm\frac{1}{2}$ , $\pm\frac{7}{2}$ .
Simplify Roots List: Simplify the list of possible rational roots: $-1$ , $1$ , $-7$ , $7$ , $-\frac{1}{2}$ , $\frac{1}{2}$ , $-\frac{7}{2}$ , $\frac{7}{2}$ .
Match with Choices: Match the simplified list of possible rational roots with the given choices: $(A) -1, (B) -\frac{1}{2}, (C) -7, (D) \frac{7}{3}$ .
Identify Possible Roots: Identify which of the possible rational roots are in the given choices: $-1$ (A), $-\frac{1}{2}$ (B), $-7$ (C). The choice (D) $\frac{7}{3}$ is not a possible root because $3$ is not a factor of the leading coefficient $2$ .

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