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Answer :
Sure! Let's tackle the synthetic division step by step to find the quotient of the polynomial when divided by [tex]\( x + 1 \)[/tex].
We're given the polynomial represented by the coefficients [tex]\([2, 7, 5]\)[/tex], which corresponds to [tex]\(2x^2 + 7x + 5\)[/tex]. We want to divide this polynomial by [tex]\(x + 1\)[/tex], or equivalently, use a divisor of [tex]\(-1\)[/tex] in synthetic division.
Steps for Synthetic Division:
1. Write down the coefficients of the polynomial to be divided, which are [tex]\(2, 7, 5\)[/tex].
2. Use the root of the divisor [tex]\(x + 1 = 0\)[/tex], which is [tex]\(-1\)[/tex], as the divisor for synthetic division.
3. Perform the division:
- First, bring down the leading coefficient, which is [tex]\(2\)[/tex].
- Next, multiply the divisor [tex]\(-1\)[/tex] by the number just written down [tex]\(2\)[/tex] and write the result [tex]\(-2\)[/tex] under the next coefficient.
- Add the result to the next coefficient [tex]\(7 + (-2) = 5\)[/tex]. Write this result below.
- Continue with the same process: multiply [tex]\(-1\)[/tex] by [tex]\(5\)[/tex] (which is the new number under the second column), resulting in [tex]\(-5\)[/tex].
- Add [tex]\(-5\)[/tex] to the last coefficient [tex]\(5 + (-5) = 0\)[/tex].
This process gives us the numbers: [tex]\(2, 5\)[/tex] followed by a remainder of [tex]\(0\)[/tex].
4. Form the quotient:
The results of synthetic division, [tex]\([2, 5]\)[/tex], represent the coefficients of the quotient polynomial. Therefore, the quotient is:
[tex]\[ 2x + 5 \][/tex]
The remainder is [tex]\(0\)[/tex], meaning [tex]\(x + 1\)[/tex] divides evenly into [tex]\(2x^2 + 7x + 5\)[/tex].
Therefore, the quotient in polynomial form is [tex]\(2x + 5\)[/tex].
The correct answer is: D. [tex]\(2x + 5\)[/tex].
We're given the polynomial represented by the coefficients [tex]\([2, 7, 5]\)[/tex], which corresponds to [tex]\(2x^2 + 7x + 5\)[/tex]. We want to divide this polynomial by [tex]\(x + 1\)[/tex], or equivalently, use a divisor of [tex]\(-1\)[/tex] in synthetic division.
Steps for Synthetic Division:
1. Write down the coefficients of the polynomial to be divided, which are [tex]\(2, 7, 5\)[/tex].
2. Use the root of the divisor [tex]\(x + 1 = 0\)[/tex], which is [tex]\(-1\)[/tex], as the divisor for synthetic division.
3. Perform the division:
- First, bring down the leading coefficient, which is [tex]\(2\)[/tex].
- Next, multiply the divisor [tex]\(-1\)[/tex] by the number just written down [tex]\(2\)[/tex] and write the result [tex]\(-2\)[/tex] under the next coefficient.
- Add the result to the next coefficient [tex]\(7 + (-2) = 5\)[/tex]. Write this result below.
- Continue with the same process: multiply [tex]\(-1\)[/tex] by [tex]\(5\)[/tex] (which is the new number under the second column), resulting in [tex]\(-5\)[/tex].
- Add [tex]\(-5\)[/tex] to the last coefficient [tex]\(5 + (-5) = 0\)[/tex].
This process gives us the numbers: [tex]\(2, 5\)[/tex] followed by a remainder of [tex]\(0\)[/tex].
4. Form the quotient:
The results of synthetic division, [tex]\([2, 5]\)[/tex], represent the coefficients of the quotient polynomial. Therefore, the quotient is:
[tex]\[ 2x + 5 \][/tex]
The remainder is [tex]\(0\)[/tex], meaning [tex]\(x + 1\)[/tex] divides evenly into [tex]\(2x^2 + 7x + 5\)[/tex].
Therefore, the quotient in polynomial form is [tex]\(2x + 5\)[/tex].
The correct answer is: D. [tex]\(2x + 5\)[/tex].
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