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Answer :
To solve the synthetic division problem, follow these steps:
1. Set up the problem:
- We are asked to divide the polynomial [tex]\(2x^2 + 7x + 5\)[/tex] by [tex]\(x + 1\)[/tex].
2. Identify the coefficients of the polynomial:
- The coefficients of [tex]\(2x^2 + 7x + 5\)[/tex] are [tex]\(2, 7, 5\)[/tex].
3. Set up synthetic division:
- The divisor is [tex]\(x + 1\)[/tex]. We use [tex]\(-1\)[/tex] (the opposite sign of the constant term in the divisor) as the number for synthetic division.
4. Perform synthetic division:
[tex]\[
\begin{array}{r|rrrr}
-1 & 2 & 7 & 5 \\
\hline
& & -2 & -5 \\
\hline
& 2 & 5 & 0 \\
\end{array}
\][/tex]
- Bring down the first coefficient [tex]\(2\)[/tex].
- Multiply [tex]\(-1\)[/tex] with [tex]\(2\)[/tex] (the number just written below), placing the result [tex]\(-2\)[/tex] under the next coefficient.
- Add [tex]\(-2\)[/tex] and [tex]\(7\)[/tex] (the second coefficient), producing [tex]\(5\)[/tex].
- Multiply [tex]\(-1\)[/tex] with [tex]\(5\)[/tex] (the new second number in the bottom row), writing [tex]\(-5\)[/tex] under the next coefficient.
- Add [tex]\(-5\)[/tex] and [tex]\(5\)[/tex] (the third coefficient), which yields a remainder of [tex]\(0\)[/tex].
5. Form the quotient:
- The result from the synthetic division gives the quotient coefficients. Here, the quotient is [tex]\(2x + 5\)[/tex].
6. Verify and choose the correct option:
- Based on the synthetic division process, the quotient of the polynomial division is [tex]\(2x + 5\)[/tex].
Therefore, the correct answer is:
C. [tex]\(2x + 5\)[/tex]
1. Set up the problem:
- We are asked to divide the polynomial [tex]\(2x^2 + 7x + 5\)[/tex] by [tex]\(x + 1\)[/tex].
2. Identify the coefficients of the polynomial:
- The coefficients of [tex]\(2x^2 + 7x + 5\)[/tex] are [tex]\(2, 7, 5\)[/tex].
3. Set up synthetic division:
- The divisor is [tex]\(x + 1\)[/tex]. We use [tex]\(-1\)[/tex] (the opposite sign of the constant term in the divisor) as the number for synthetic division.
4. Perform synthetic division:
[tex]\[
\begin{array}{r|rrrr}
-1 & 2 & 7 & 5 \\
\hline
& & -2 & -5 \\
\hline
& 2 & 5 & 0 \\
\end{array}
\][/tex]
- Bring down the first coefficient [tex]\(2\)[/tex].
- Multiply [tex]\(-1\)[/tex] with [tex]\(2\)[/tex] (the number just written below), placing the result [tex]\(-2\)[/tex] under the next coefficient.
- Add [tex]\(-2\)[/tex] and [tex]\(7\)[/tex] (the second coefficient), producing [tex]\(5\)[/tex].
- Multiply [tex]\(-1\)[/tex] with [tex]\(5\)[/tex] (the new second number in the bottom row), writing [tex]\(-5\)[/tex] under the next coefficient.
- Add [tex]\(-5\)[/tex] and [tex]\(5\)[/tex] (the third coefficient), which yields a remainder of [tex]\(0\)[/tex].
5. Form the quotient:
- The result from the synthetic division gives the quotient coefficients. Here, the quotient is [tex]\(2x + 5\)[/tex].
6. Verify and choose the correct option:
- Based on the synthetic division process, the quotient of the polynomial division is [tex]\(2x + 5\)[/tex].
Therefore, the correct answer is:
C. [tex]\(2x + 5\)[/tex]
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