Questions: When dividing the polynomial (x^3+4 x^2+x-6) by (x-2) using synthetic division, which of the following setup boxes would be used? a.) (2 mid 321-6) b.) (2 1 4 1 -6) d.) (-2 4 1 -6)

When dividing the polynomial (x^3+4 x^2+x-6) by (x-2) using synthetic division, which of the following setup boxes would be used? a.) (2 mid 321-6)
b.) (2 1 4 1 -6)
d.) (-2 4 1 -6)

Transcript text: When dividing the polynomial $x^{3}+4 x^{2}+x-6$ by $x-2$ using synthetic division, which of the following setup boxes would be used? a.) $2 \mid 321-6$ b.) $2 \left\lvert\, \begin{array}{llll}1 & 4 & 1 & -6\end{array}\right.$ $\qquad$ d.) $-2 \left\lvert\, \begin{array}{lllll} & 4 & 1 & -6\end{array}\right.$

Solution

Solution Steps

To determine the correct setup for synthetic division, we need to identify the coefficients of the polynomial $x^3 + 4x^2 + x - 6$ and the divisor $x - 2$. The coefficients are [1, 4, 1, -6], and the divisor is 2. The correct setup for synthetic division involves placing the divisor outside the box and the coefficients inside the box.

Step 1: Identify the Coefficients and Divisor

The polynomial given is $x^3 + 4x^2 + x - 6$. The coefficients of this polynomial are $[1, 4, 1, -6]$. The divisor is $x - 2$, which means the value to use in synthetic division is $2$.

Step 2: Set Up Synthetic Division

In synthetic division, we place the divisor outside the box and the coefficients inside the box. The setup for synthetic division is: \[ 2 \mid 1 \quad 4 \quad 1 \quad -6 \]