To solve the problem using synthetic division, we'll follow a step-by-step approach to divide the polynomial [tex]\(2x^2 + 5x - 7\)[/tex] by [tex]\(x-2\)[/tex].
Step 1: Set up the synthetic division.
We'll use the zeros of [tex]\(x-2\)[/tex], which is [tex]\(2\)[/tex]. Set up the coefficients of the polynomial: [tex]\(2, 5, -7\)[/tex].
Write down the coefficients:
[tex]\[ \begin{array}{r|rrr}
2 & 2 & 5 & -7 \\
\end{array} \][/tex]
Step 2: Perform the synthetic division.
1. Bring down the leading coefficient [tex]\(2\)[/tex].
[tex]\[ \begin{array}{r|rrr}
2 & 2 & 5 & -7 \\
& & & \\
& 2 & & \\
\end{array} \][/tex]
2. Multiply [tex]\(2\)[/tex] by the leading coefficient of [tex]\(x-2\)[/tex], which is [tex]\(2\)[/tex], and place it under the next coefficient:
[tex]\[ \begin{array}{r|rrr}
2 & 2 & 5 & -7 \\
& & 4 & \\
& 2 & & \\
\end{array} \][/tex]
3. Add the values in the second column: [tex]\(5 + 4 = 9\)[/tex].
[tex]\[ \begin{array}{r|rrr}
2 & 2 & 5 & -7 \\
& & 4 & 18 \\
& 2 & 9 & \\
\end{array} \][/tex]
4. Repeat the multiplication and addition with the last coefficient.
[tex]\[ \begin{array}{r|rrr}
2 & 2 & 5 & -7 \\
& & 4 & 18 \\
& 2 & 9 & 11 \\
\end{array} \][/tex]
We performed [tex]\(2 \times 9\)[/tex] and placed it below the remaining coefficients, then added up [tex]\( -7 + 18 = 11\)[/tex].
Step 3: Interpret the result.
Our quotient from the synthetic division is [tex]\(2x + 9\)[/tex] and the remainder is [tex]\(11\)[/tex].
Final Answer:
The quotient is [tex]\(\boxed{2x + 9 + \frac{11}{x-2}}.\)[/tex]
Therefore, the correct answer is:
[tex]\[ \text{B. } 2 x + 9 + \frac{11}{x - 2} \][/tex]
