Question

which is the correct first line for dividing the function $f(x)=x^3 - 12x + 16$ by $(x - 2)$ using synthetic division?
(1 point)
$\bigcirc\\ \

$$\begin{array}{r|rrr}2 & 1 & -12 & 16\\\\\\hline\\end{array}$$

$
$\bigcirc\\ \

$$\begin{array}{r|rrr}-2 & 1 & -12 & 16\\\\\\hline\\end{array}$$

$
$\bigcirc\\ \

$$\begin{array}{r|rrr}2 & 1 & 0 & -12 & 16\\\\\\hline\\end{array}$$

$
$\bigcirc\\ \

$$\begin{array}{r|rrr}-2 & 1 & 0 & -12 & 16\\\\\\hline\\end{array}$$

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Explanation:

Step1: Recall synthetic division rules

For dividing a polynomial \( f(x) = a_nx^n + a_{n - 1}x^{n - 1}+\dots+a_1x + a_0 \) by \( (x - c) \), the first line of synthetic division uses the root \( c \) on the left and the coefficients of \( f(x) \) (including zeros for missing terms) on the right.

Step2: Analyze the given polynomial

The polynomial is \( f(x)=x^3 - 12x + 16 \). Rewriting it in standard form (descending powers of \( x \)): \( f(x)=x^3+0x^2 - 12x + 16 \). So the coefficients are \( 1 \) (for \( x^3 \)), \( 0 \) (for \( x^2 \)), \( - 12 \) (for \( x \)), and \( 16 \) (constant term).

Step3: Determine the root \( c \)

We are dividing by \( (x - 2) \), so by the factor theorem, the root \( c \) is \( 2 \) (since \( x - 2=0\implies x = 2 \)).

Step4: Form the first line

Using the root \( c = 2 \) and the coefficients \( 1,0,-12,16 \), the first line of synthetic division is \( 2\mid1\ 0\ - 12\ 16 \).

Answer:

\( 2\mid1\ 0\ - 12\ 16 \) (the third option: \( \boldsymbol{2\mid1\ 0\ -12\ 16} \))