Which synthetic division shows how to find the quotient?x5−4x3+2x+6÷x+2 (1 point)ResponsesImage with alt text: A synthetic division display with a divisor of negative 2.Image with alt text: A synthetic division display with a divisor of negative 2.Image with alt text: A synthetic division display with a divisor of negative 2.

Answer 1

To perform synthetic division for dividing \( x^5 - 4x^3 + 2x + 6 \) by \( x + 2 \), we need to use the root of the divisor \( x + 2 \), which is \( -2 \).

Here are the steps for synthetic division:

1. Set up the synthetic division using the coefficients from the polynomial \( x^5 - 4x^3 + 2x + 6 \). Since this polynomial is missing \( x^4 \), we will include a coefficient of 0 for it. Thus, the coefficients are:

   \[ 1, 0, -4, 0, 2, 6 \]

2. Perform synthetic division using -2 as follows:

   • Write -2 to the left and the coefficients to the right.
   • Drop down the leading coefficient (1).
   • Multiply -2 by the number just dropped down and write the result under the next coefficient.
   • Add the values in the column.

Here’s how the setup looks:

-2 | 1   0   -4   0   2   6
   |     -2    4  -8  16 -32
   -------------------------
     1  -2   0   -8   18 -26

The rows are calculated as follows:

1. Drop down the 1.
2. Calculate \( -2 \cdot 1 = -2 \), then add \( 0 - 2 = -2 \).
3. Calculate \( -2 \cdot -2 = 4 \), then add \( -4 + 4 = 0 \).
4. Calculate \( -2 \cdot 0 = 0 \), then add \( 0 + 0 = 0 \).
5. Calculate \( -2 \cdot 0 = 0 \), then add \( 2 + 0 = 2 \).
6. Calculate \( -2 \cdot 2 = -4 \), then finally add \( 6 - 4 = 2 \).

So, we end with the row \( 1, -2, 0, -8, 2 \) and a remainder of \( 2 \).

This means the quotient is \( x^4 - 2x^3 + 0x^2 - 8x + 2 \) with a remainder of 2.

Thus, the answer to your question regarding the synthetic division for dividing \( x^5 - 4x^3 + 2x + 6 \) by \( x + 2 \) is represented in the correct synthetic division display with a divisor of \( -2 \).