Answer :
To solve the synthetic division problem, let's consider the polynomial division of [tex]\(2x^2 + 7x + 5\)[/tex] by [tex]\(x + 1\)[/tex] using synthetic division.
1. Set up the synthetic division:
- The divisor here is [tex]\(x + 1\)[/tex], which means our root (or zero) is [tex]\(-1\)[/tex].
- Write down the coefficients of the polynomial: [tex]\([2, 7, 5]\)[/tex].
2. Perform synthetic division:
- Bring down the first coefficient, 2.
- Multiply it by the root [tex]\(-1\)[/tex] and add it to the next coefficient:
- [tex]\(2 \times (-1) + 7 = 5\)[/tex].
- Write down the result, 5.
- Multiply 5 by [tex]\(-1\)[/tex] and add it to the next coefficient:
- [tex]\(5 \times (-1) + 5 = 0\)[/tex].
- Write down the result, 0.
3. Interpret the results:
- The numbers obtained after division represent the coefficients of the quotient polynomial.
- The final row of numbers is [tex]\([2, 5]\)[/tex], and the remainder is 0.
4. Write the quotient in polynomial form:
- The quotient polynomial is [tex]\(2x + 5\)[/tex].
So, the quotient of the division is [tex]\(2x + 5\)[/tex], which matches option D.
1. Set up the synthetic division:
- The divisor here is [tex]\(x + 1\)[/tex], which means our root (or zero) is [tex]\(-1\)[/tex].
- Write down the coefficients of the polynomial: [tex]\([2, 7, 5]\)[/tex].
2. Perform synthetic division:
- Bring down the first coefficient, 2.
- Multiply it by the root [tex]\(-1\)[/tex] and add it to the next coefficient:
- [tex]\(2 \times (-1) + 7 = 5\)[/tex].
- Write down the result, 5.
- Multiply 5 by [tex]\(-1\)[/tex] and add it to the next coefficient:
- [tex]\(5 \times (-1) + 5 = 0\)[/tex].
- Write down the result, 0.
3. Interpret the results:
- The numbers obtained after division represent the coefficients of the quotient polynomial.
- The final row of numbers is [tex]\([2, 5]\)[/tex], and the remainder is 0.
4. Write the quotient in polynomial form:
- The quotient polynomial is [tex]\(2x + 5\)[/tex].
So, the quotient of the division is [tex]\(2x + 5\)[/tex], which matches option D.
