To express the given division problem [tex]\( \frac{6x^2 - x + 5}{x + 7} \)[/tex] using synthetic division, follow these steps:
1. Identify the coefficients of the polynomial: For the polynomial [tex]\( 6x^2 - x + 5 \)[/tex], the coefficients are [tex]\( 6 \)[/tex], [tex]\( -1 \)[/tex], and [tex]\( 5 \)[/tex].
2. Set up the synthetic division:
- In synthetic division, we need to use the opposite sign of the constant term from the divisor. Here, the divisor is [tex]\( x + 7 \)[/tex], so the constant term is [tex]\( 7 \)[/tex]. Therefore, we will use [tex]\( -7 \)[/tex].
3. Place the coefficients in a row:
- Write the coefficients of the polynomial [tex]\( 6 \)[/tex], [tex]\( -1 \)[/tex], and [tex]\( 5 \)[/tex] in a row.
4. Combine the opposite of the divisor's constant with the row of coefficients:
- The setup should look like this:
[tex]\[
-7 \longdiv { 6 \quad -1 \quad 5 }
\][/tex]
Thus, the correct synthetic division form is:
[tex]\[ -7 \longdiv { 6 \quad -1 \quad 5 } \][/tex]
Therefore, the correct choice is:
[tex]\[ \text{D. } - 7 \longdiv { 6 - 1 \quad 5 } \][/tex]
