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Answer :
To solve this synthetic division problem, you are dividing the polynomial [tex]\(2x^2 + 7x + 5\)[/tex] by the binomial [tex]\(x + 1\)[/tex].
Here’s a step-by-step explanation of how the synthetic division works for this:
1. Identify the coefficients: The polynomial [tex]\(2x^2 + 7x + 5\)[/tex] has the coefficients [2, 7, 5].
2. Set up synthetic division: The divisor [tex]\(x + 1\)[/tex] means you will use -1 for synthetic division. This is because we use the root of the divisor, which is -1 in this case (since [tex]\(x + 1 = x - (-1)\)[/tex]).
3. Start the synthetic division process:
- Write -1 on the left and the coefficients [2, 7, 5] on the right.
- Bring down the first coefficient, which is 2.
4. Perform the process:
- Multiply -1 by the first coefficient 2 and write the result under the next coefficient:
[tex]\[
-1 \times 2 = -2
\][/tex]
- Add this result to 7 (the next coefficient):
[tex]\[
7 + (-2) = 5
\][/tex]
- Write 5 below the line.
- Multiply -1 by 5 (the new number just written) and write under the next coefficient:
[tex]\[
-1 \times 5 = -5
\][/tex]
- Add this to 5 (the last coefficient):
[tex]\[
5 + (-5) = 0
\][/tex]
- Write 0 below the line, which confirms there's a remainder of 0.
5. Form the quotient: The resulting numbers above the line, [2, 5], give you the coefficients of the quotient polynomial. The quotient of the division [tex]\(2x^2 + 7x + 5\)[/tex] by [tex]\(x + 1\)[/tex] is [tex]\(2x + 5\)[/tex].
Therefore, the polynomial quotient is [tex]\(2x + 5\)[/tex].
The correct answer is:
A. [tex]\(2x + 5\)[/tex]
Here’s a step-by-step explanation of how the synthetic division works for this:
1. Identify the coefficients: The polynomial [tex]\(2x^2 + 7x + 5\)[/tex] has the coefficients [2, 7, 5].
2. Set up synthetic division: The divisor [tex]\(x + 1\)[/tex] means you will use -1 for synthetic division. This is because we use the root of the divisor, which is -1 in this case (since [tex]\(x + 1 = x - (-1)\)[/tex]).
3. Start the synthetic division process:
- Write -1 on the left and the coefficients [2, 7, 5] on the right.
- Bring down the first coefficient, which is 2.
4. Perform the process:
- Multiply -1 by the first coefficient 2 and write the result under the next coefficient:
[tex]\[
-1 \times 2 = -2
\][/tex]
- Add this result to 7 (the next coefficient):
[tex]\[
7 + (-2) = 5
\][/tex]
- Write 5 below the line.
- Multiply -1 by 5 (the new number just written) and write under the next coefficient:
[tex]\[
-1 \times 5 = -5
\][/tex]
- Add this to 5 (the last coefficient):
[tex]\[
5 + (-5) = 0
\][/tex]
- Write 0 below the line, which confirms there's a remainder of 0.
5. Form the quotient: The resulting numbers above the line, [2, 5], give you the coefficients of the quotient polynomial. The quotient of the division [tex]\(2x^2 + 7x + 5\)[/tex] by [tex]\(x + 1\)[/tex] is [tex]\(2x + 5\)[/tex].
Therefore, the polynomial quotient is [tex]\(2x + 5\)[/tex].
The correct answer is:
A. [tex]\(2x + 5\)[/tex]
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