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A polynomial function $f(x)$ with integer coefficients has a leading coefficient of $2$ 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) $\frac{4}{3}$ $\newline$ (B) $\frac{1}{2}$ $\newline$ (C) $1$ $\newline$ (D) $\frac{1}{10}$

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Rational root theorem

Full solution

Q. A polynomial function

f(x)

with integer coefficients has a leading coefficient of

2

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)

\frac{4}{3}

\newline

(B)

\frac{1}{2}

\newline

(C)

1

\newline

(D)

\frac{1}{10}

Understand Rational Root Theorem: The Rational Root Theorem states that any rational root, expressed in its lowest terms $\frac{p}{q}$ , must have $p$ as a factor of the constant term and $q$ as a factor of the leading coefficient.
List Factors of Constant Term: List the factors of the constant term, which is $1$ : $\pm 1$ .
List Factors of Leading Coefficient: List the factors of the leading coefficient, which is $2$ : $\pm 1$ , $\pm 2$ .
Form Possible Fractions: Form all possible fractions $p/q$ using the factors of the constant term for $p$ and the factors of the leading coefficient for $q$ : $\pm 1/1$ , $\pm 1/2$ .
Simplify Fractions: Simplify the fractions to get the possible rational roots: $\pm 1, \pm \frac{1}{2}$ .
Check Given Options: Check the given options against the possible rational roots: (A) $\frac{4}{3}$ is not a possible root because $3$ is not a factor of the leading coefficient. (B) $\frac{1}{2}$ is a possible root. (C) $1$ is a possible root. (D) $\frac{1}{10}$ is not a possible root because $10$ is not a factor of the leading coefficient.

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