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A polynomial function $f(x)$ with integer coefficients has a leading coefficient of $-5$ and a constant term of $2$ . According to the Rational Root Theorem, which of the following are possible roots of $f(x)$ ? $\newline$ Multi-select Choices: $\newline$ (A) $-\frac{1}{5}$ $\newline$ (B) $\frac{2}{17}$ $\newline$ (C) $2$ $\newline$ (D) $\frac{10}{3}$

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Rational root theorem

Full solution

Q. A polynomial function

f(x)

with integer coefficients has a leading coefficient of

-5

and a constant term of

2

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

f(x)

?

\newline

Multi-select Choices:

\newline

(A)

-\frac{1}{5}

\newline

(B)

\frac{2}{17}

\newline

(C)

2

\newline

(D)

\frac{10}{3}

Understand Rational Root Theorem: The Rational Root Theorem states that any rational root, expressed in its simplest form $\frac{p}{q}$ , must have $p$ as a factor of the constant term and $q$ as a factor of the leading coefficient.
Factor Constant Term: List the factors of the constant term $2$ : $\pm1$ , $\pm2$ .
Factor Leading Coefficient: List the factors of the leading coefficient $-5$ : $\pm 1$ , $\pm 5$ .
Generate Possible Rational Roots: Generate the possible rational roots by combining the factors of the constant term with the factors of the leading coefficient: $\pm\frac{1}{1}$ , $\pm\frac{1}{5}$ , $\pm\frac{2}{1}$ , $\pm\frac{2}{5}$ .
Simplify Roots List: Simplify the list of possible rational roots: $-1$ , $-\frac{1}{5}$ , $1$ , $\frac{1}{5}$ , $-2$ , $-\frac{2}{5}$ , $2$ , $\frac{2}{5}$ .
Check Given Options: Check the given options against the list of possible rational roots. $\newline$ (A) $-\frac{1}{5}$ is in the list, so it's a possible root. $\newline$ (B) $\frac{2}{17}$ is not in the list because $17$ is not a factor of the leading coefficient $-5$ . $\newline$ (C) $2$ is in the list, so it's a possible root. $\newline$ (D) $\frac{10}{3}$ is not in the list because $3$ is not a factor of the leading coefficient $-5$ .

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