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Answer :
To solve the synthetic division problem, we need to divide the polynomial represented by the coefficients [tex]\(2, 7, 5\)[/tex] by [tex]\(x + 1\)[/tex], which implies using [tex]\(-1\)[/tex] as the number for synthetic division.
Here's how we can do it step-by-step:
1. Set Up Synthetic Division:
- Write down the coefficients of the polynomial: [tex]\(2, 7, 5\)[/tex].
- We'll use [tex]\(-1\)[/tex] (the root of the divisor [tex]\(x + 1 = 0\)[/tex]) for the synthetic division process.
2. Perform Synthetic Division:
- Bring down the first coefficient [tex]\(2\)[/tex] directly as it is.
- Multiply the root [tex]\(-1\)[/tex] by this [tex]\(2\)[/tex] to get [tex]\(-2\)[/tex].
- Add this [tex]\(-2\)[/tex] to the next coefficient [tex]\(7\)[/tex] to get [tex]\(5\)[/tex].
- Multiply [tex]\(-1\)[/tex] by the new number [tex]\(5\)[/tex] to get [tex]\(-5\)[/tex].
- Add this [tex]\(-5\)[/tex] to the last coefficient [tex]\(5\)[/tex] to get [tex]\(0\)[/tex], which is the remainder.
3. Find the Quotient:
- The result from the synthetic division gives us the coefficients of the quotient polynomial.
- The coefficients are [tex]\(2\)[/tex] for [tex]\(x\)[/tex] and [tex]\(5\)[/tex] for the constant term. So the quotient is [tex]\(2x + 5\)[/tex].
Since the remainder is [tex]\(0\)[/tex], there is no remainder, and the division is exact.
Thus, the quotient in polynomial form is [tex]\(2x + 5\)[/tex]. Therefore, the correct answer is option D: [tex]\(2x + 5\)[/tex].
Here's how we can do it step-by-step:
1. Set Up Synthetic Division:
- Write down the coefficients of the polynomial: [tex]\(2, 7, 5\)[/tex].
- We'll use [tex]\(-1\)[/tex] (the root of the divisor [tex]\(x + 1 = 0\)[/tex]) for the synthetic division process.
2. Perform Synthetic Division:
- Bring down the first coefficient [tex]\(2\)[/tex] directly as it is.
- Multiply the root [tex]\(-1\)[/tex] by this [tex]\(2\)[/tex] to get [tex]\(-2\)[/tex].
- Add this [tex]\(-2\)[/tex] to the next coefficient [tex]\(7\)[/tex] to get [tex]\(5\)[/tex].
- Multiply [tex]\(-1\)[/tex] by the new number [tex]\(5\)[/tex] to get [tex]\(-5\)[/tex].
- Add this [tex]\(-5\)[/tex] to the last coefficient [tex]\(5\)[/tex] to get [tex]\(0\)[/tex], which is the remainder.
3. Find the Quotient:
- The result from the synthetic division gives us the coefficients of the quotient polynomial.
- The coefficients are [tex]\(2\)[/tex] for [tex]\(x\)[/tex] and [tex]\(5\)[/tex] for the constant term. So the quotient is [tex]\(2x + 5\)[/tex].
Since the remainder is [tex]\(0\)[/tex], there is no remainder, and the division is exact.
Thus, the quotient in polynomial form is [tex]\(2x + 5\)[/tex]. Therefore, the correct answer is option D: [tex]\(2x + 5\)[/tex].
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