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A polynomial function $f(x)$ with integer coefficients has a leading coefficient of $3$ and a constant term of $1$ . According to the Rational Root Theorem, which of the following are possible roots of $f(x)$ ? $\newline$ Multi-select Choices: $\newline$ (A) $1$ $\newline$ (B) $-\frac{1}{3}$ $\newline$ (C) $\frac{1}{3}$ $\newline$ (D) $-19$

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Rational root theorem

Full solution

Q. A polynomial function

f(x)

with integer coefficients has a leading coefficient of

3

and a constant term of

1

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

f(x)

?

\newline

Multi-select Choices:

\newline

(A)

1

\newline

(B)

-\frac{1}{3}

\newline

(C)

\frac{1}{3}

\newline

(D)

-19

List Factors Constant Term: List all possible factors of the constant term, which is $1$ . The factors of $1$ are $\pm1$ .
List Factors Leading Coefficient: List all possible factors of the leading coefficient, which is $3$ . The factors of $3$ are $\pm1$ and $\pm3$ .
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, we get $\pm\frac{1}{1}$ , $\pm\frac{1}{3}$ .
Simplify Roots: Simplify the possible rational roots. We have $1$ , $-1$ , $\frac{1}{3}$ , and $-\frac{1}{3}$ .
Check Given Choices: Check the given choices against the possible roots we found. Choices (A) $1$ , (B) $-\frac{1}{3}$ , and (C) $\frac{1}{3}$ are the possible roots. Choice (D) $-19$ is not a possible root because $19$ is not a factor of the constant term $1$ .

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