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Answer :
To solve the synthetic division problem, let's break it down into simple steps:
1. Identify the Dividend and Divisor:
- We're given the polynomial coefficients for the dividend: [tex]\(2, 7, 5\)[/tex]. This corresponds to the polynomial [tex]\(2x^2 + 7x + 5\)[/tex].
- The divisor is [tex]\(x + 1\)[/tex], and we need the root for synthetic division, which is [tex]\(-1\)[/tex].
2. Set Up for Synthetic Division:
- Write down the coefficients of the dividend: [tex]\([2, 7, 5]\)[/tex].
- The root from the divisor [tex]\(x + 1 = 0\)[/tex] gives [tex]\(-1\)[/tex].
3. Perform Synthetic Division:
a. Bring Down the Leading Coefficient:
- Start by carrying down the first coefficient, [tex]\(2\)[/tex].
b. Multiply and Add:
- Multiply this number by the root [tex]\(-1\)[/tex], giving [tex]\(-2\)[/tex].
- Add this result to the next coefficient [tex]\(7\)[/tex]. The calculation is [tex]\(7 + (-2) = 5\)[/tex].
c. Continue the Process:
- Now take the [tex]\(5\)[/tex] (from the previous step), multiply by the root [tex]\(-1\)[/tex], which gives [tex]\(-5\)[/tex].
- Add this to the last coefficient [tex]\(5\)[/tex]. The calculation is [tex]\(5 + (-5) = 0\)[/tex].
4. Result of Synthetic Division:
- The synthetic division gives us new coefficients [tex]\([2, 5]\)[/tex] for the quotient, and a remainder of [tex]\(0\)[/tex].
5. Write the Quotient in Polynomial Form:
- The quotient is obtained from the coefficients [tex]\([2, 5]\)[/tex].
- The polynomial from these coefficients is [tex]\(2x + 5\)[/tex].
Hence, the quotient in polynomial form when dividing by [tex]\(x + 1\)[/tex] is:
A. [tex]\(2x + 5\)[/tex]
1. Identify the Dividend and Divisor:
- We're given the polynomial coefficients for the dividend: [tex]\(2, 7, 5\)[/tex]. This corresponds to the polynomial [tex]\(2x^2 + 7x + 5\)[/tex].
- The divisor is [tex]\(x + 1\)[/tex], and we need the root for synthetic division, which is [tex]\(-1\)[/tex].
2. Set Up for Synthetic Division:
- Write down the coefficients of the dividend: [tex]\([2, 7, 5]\)[/tex].
- The root from the divisor [tex]\(x + 1 = 0\)[/tex] gives [tex]\(-1\)[/tex].
3. Perform Synthetic Division:
a. Bring Down the Leading Coefficient:
- Start by carrying down the first coefficient, [tex]\(2\)[/tex].
b. Multiply and Add:
- Multiply this number by the root [tex]\(-1\)[/tex], giving [tex]\(-2\)[/tex].
- Add this result to the next coefficient [tex]\(7\)[/tex]. The calculation is [tex]\(7 + (-2) = 5\)[/tex].
c. Continue the Process:
- Now take the [tex]\(5\)[/tex] (from the previous step), multiply by the root [tex]\(-1\)[/tex], which gives [tex]\(-5\)[/tex].
- Add this to the last coefficient [tex]\(5\)[/tex]. The calculation is [tex]\(5 + (-5) = 0\)[/tex].
4. Result of Synthetic Division:
- The synthetic division gives us new coefficients [tex]\([2, 5]\)[/tex] for the quotient, and a remainder of [tex]\(0\)[/tex].
5. Write the Quotient in Polynomial Form:
- The quotient is obtained from the coefficients [tex]\([2, 5]\)[/tex].
- The polynomial from these coefficients is [tex]\(2x + 5\)[/tex].
Hence, the quotient in polynomial form when dividing by [tex]\(x + 1\)[/tex] is:
A. [tex]\(2x + 5\)[/tex]
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