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A polynomial function $h(x)$ with integer coefficients has a leading coefficient of $16$ and a constant term of $-1$ . According to the Rational Root Theorem, which of the following are possible roots of $h(x)$ ? $\newline$ Multi-select Choices: $\newline$ (A) $\frac{18}{5}$ $\newline$ (B) $-18$ $\newline$ (C) $7$ $\newline$ (D) $1$

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Rational root theorem

Full solution

Q. A polynomial function

h(x)

with integer coefficients has a leading coefficient of

16

and a constant term of

-1

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

h(x)

?

\newline

Multi-select Choices:

\newline

(A)

\frac{18}{5}

\newline

(B)

-18

\newline

(C)

7

\newline

(D)

1

The 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 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 $-1$ , which are $\pm 1$ .
Leading Coefficient Factors: List the factors of the leading coefficient $16$ , which are $\pm1$ , $\pm2$ , $\pm4$ , $\pm8$ , and $\pm16$ .
Possible Rational Roots: According to the Rational Root Theorem, the possible rational roots are the factors of the constant term divided by the factors of the leading coefficient. So, the possible roots are $\pm\frac{1}{1}$ , $\pm\frac{1}{2}$ , $\pm\frac{1}{4}$ , $\pm\frac{1}{8}$ , and $\pm\frac{1}{16}$ .
Check Given Options: Check each of the given options against the possible roots: $\newline$ (A) $\frac{18}{5}$ is not a possible root because $5$ is not a factor of $16$ . $\newline$ (B) $-18$ is not a possible root because $18$ is not a factor of $-1$ . $\newline$ (C) $7$ is not a possible root because $7$ is not a factor of $-1$ . $\newline$ (D) $1$ is a possible root because it is $5$ $0$ , and both $1$ and $1$ are factors of $-1$ and $16$ , respectively.

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