To perform synthetic division for the polynomial \( 8x^3 + 6x^2 - 18x + 4 \) by the divisor \( 4x - 1 \), we first need to rewrite the divisor in the form \( x - c \).
Thus, we have:
\( 4x - 1 = 0 \implies x = \frac{1}{4} \)
In synthetic division, we will use \( \frac{1}{4} \) as our value. Now let's set up the synthetic division:
- Write down the coefficients of the polynomial: \( 8, 6, -18, 4 \).
- Use \( \frac{1}{4} \) for synthetic division.
The process involves bringing down the leading coefficient, multiplying by \( \frac{1}{4} \), then adding the results in a column fashion.
From the options provided, we need to match the process with the correct representation:
Among the choices, the correct synthetic division setup should show:
- The first row of coefficients.
- The row after the calculations.
- The last row showing the results.
Based on the descriptions provided, if the synthetic division leads to zero remainder, you would expect a resulting polynomial and a zero in the last position of the bottom row.
The option that describes a final output of coefficients properly aligns with the setup of synthetic division we would expect.
Based on the provided choices, the correct synthetic division display corresponds to:
A synthetic division display with a divisor of \( \frac{1}{4} \). The first row has entries of 2, \( \frac{6}{4} \), \( -\frac{18}{4} \), and 1. The second row has entries of \( \frac{2}{4} \), \( \frac{2}{4} \), and negative 1. The bottom row has entries of 2, 2, -4, and 0.
This option correctly reflects the setup for synthetic division with the divisor and demonstrates the calculations involved.
1 answer