Which synthetic division shows how to find the quotient?8x3+6x2−18x+4÷4x−1 (1 point)ResponsesImage with alt text: A synthetic division display with a divisor of 1 fourth. The first row has entries of 2, 6 fourths, negative 18 fourths, and 1. The second row has entries of 2 fourths, 2 fourths, and negative 1. The bottom row has entries of 2, 2, negative 4, and 0.Image with alt text: A synthetic division display with a divisor of 1 fourth. The first row has entries of 2, 6 fourths, 18 fourths, and 1. The second row has entries of 2 fourths, 2 fourths, and 5 fourths. The bottom row has entries of 2, 2, 5, and 9 fourths.Image with alt text: A synthetic division display with a divisor of 1 fourth.Image with alt text: A synthetic division display with a divisor of 1 fourth. The first row has entries of 2, 6 fourths, negative 18 fourths, and 1. The second row has entries of 2 fourths, 2 fourths, and 1. The bottom row has entries of 2, 2, 4, and 2.

Answer 1

To perform synthetic division of the polynomial \(8x^3 + 6x^2 - 18x + 4\) by \(4x - 1\), we first identify the value we will be using for synthetic division. The divisor \(4x - 1\) can be set to zero to find the appropriate value:

\[ 4x - 1 = 0 \Rightarrow 4x = 1 \Rightarrow x = \frac{1}{4} \]

Now we will use \(\frac{1}{4}\) in synthetic division. Here's how synthetic division works:

1. Set up the coefficients: The coefficients of the polynomial are \(8\), \(6\), \(-18\), and \(4\).

2. Perform synthetic division:

   • Write down the coefficients: \(8, 6, -18, 4\).
   • Bring down the first coefficient \(8\).
   • Multiply \(8\) by \(\frac{1}{4}\) and add it to the next coefficient \(6\).
   • Repeat this process for each coefficient.

Here’s a breakdown of each step:

• Start with: \[ \begin{array}{r|rrrr} \frac{1}{4} & 8 & 6 & -18 & 4 \ & & & & \ \hline & 8 & & & \ \end{array} \]

• Bring down the \(8\):

\[ \begin{array}{r|rrrr} \frac{1}{4} & 8 & 6 & -18 & 4 \ & & & & \ \hline & 8 & & & \ \end{array} \]

• Multiply \(8 \cdot \frac{1}{4} = 2\), then add it to \(6\): \[ \begin{array}{r|rrrr} \frac{1}{4} & 8 & 6 & -18 & 4 \ & & 2 & & \ \hline & 8 & 8 & & \ \end{array} \]

• Multiply \(8 \cdot \frac{1}{4} = 2\), and add to \(-18\): \[ \begin{array}{r|rrrr} \frac{1}{4} & 8 & 6 & -18 & 4 \ & & 2 & 2 & \ \hline & 8 & 8 & -16 & \ \end{array} \]

• Multiply \(-16 \cdot \frac{1}{4} = -4\), and add to \(4\): \[ \begin{array}{r|rrrr} \frac{1}{4} & 8 & 6 & -18 & 4 \ & & 2 & 2 & -4 \ \hline & 8 & 8 & -16 & 0 \ \end{array} \]

At the end of this process, we find that the bottom row (the result of the synthetic division) corresponds to \(8, 2, -4\) with a remainder of \(0\). Thus, the quotient is \(8x^2 + 2x - 4\).

Based on the provided options, you may identify the correct setup or output that matches this operation. The correct middle row under the divisor should include \(2\) and the bottom row should represent the coefficients of the quotient polynomial followed by a \(0\) remainder.

If you see an image that corresponds to \(8, 2, -4, 0\) in the output format, that is the correct representation of the synthetic division result.