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A polynomial function $f(x)$ with integer coefficients has a leading coefficient of $7$ and a constant term of $5$ . According to the Rational Root Theorem, which of the following are possible roots of $f(x)$ ? $\newline$ Multi-select Choices: $\newline$ (A) $-\frac{10}{17}$ $\newline$ (B) $-\frac{5}{9}$ $\newline$ (C) $-8$ $\newline$ (D) $-1$

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Rational root theorem

Full solution

Q. A polynomial function

f(x)

with integer coefficients has a leading coefficient of

7

and a constant term of

5

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

f(x)

?

\newline

Multi-select Choices:

\newline

(A)

-\frac{10}{17}

\newline

(B)

-\frac{5}{9}

\newline

(C)

-8

\newline

(D)

-1

Theorem Explanation: 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 have $p$ as a factor of the constant term and $q$ as a factor of the leading coefficient.
Constant Term Factors: List the factors of the constant term $5$ : $\pm1$ , $\pm5$ .
Leading Coefficient Factors: List the factors of the leading coefficient $7$ : $\pm1$ , $\pm7$ .
Generate Possible Roots: Generate the possible rational roots by taking all combinations of factors of the constant term over factors of the leading coefficient: $\pm\frac{1}{1}$ , $\pm\frac{5}{1}$ , $\pm\frac{1}{7}$ , $\pm\frac{5}{7}$ .
Simplify Roots List: Simplify the list of possible rational roots: $\pm 1$ , $\pm 5$ , $\pm \frac{1}{7}$ , $\pm \frac{5}{7}$ .
Check Given Choices: Check which of the given choices match the possible rational roots: (A) $-\frac{10}{17}$ is not a possible root because $17$ is not a factor of $7$ . (B) $-\frac{5}{9}$ is not a possible root because $9$ is not a factor of $7$ . (C) $-8$ is not a possible root because $8$ is not a factor of $5$ . (D) $-1$ is a possible root because it is in the list of possible rational roots.

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