Answer :
Sure, let's solve the synthetic division problem step-by-step.
The problem involves dividing the polynomial represented by its coefficients [tex]\([2, 7, 5]\)[/tex] by [tex]\(x + 1\)[/tex]. In synthetic division, the divisor is given in the form [tex]\(x - c\)[/tex]. Here, [tex]\(c\)[/tex] is [tex]\(-1\)[/tex] (since the expression under the division is [tex]\(x + 1\)[/tex]).
### Steps:
1. Set up the synthetic division:
- Write the coefficients of the polynomial: [tex]\(2, 7, 5\)[/tex].
- The number to the left of the division bar (the zero of the divisor) is [tex]\(-1\)[/tex].
2. Perform the synthetic division:
- Bring down the first coefficient (2) as it is.
- Multiply [tex]\(-1\)[/tex] by this number (2) and put the result under the next coefficient: [tex]\(-1 \times 2 = -2\)[/tex]. Write [tex]\(-2\)[/tex] below 7.
- Add the numbers in the second column: [tex]\(7 + (-2) = 5\)[/tex].
- Multiply [tex]\(-1\)[/tex] by the new result (5) and put this result under the next coefficient: [tex]\(-1 \times 5 = -5\)[/tex]. Write [tex]\(-5\)[/tex] below 5.
- Add these two numbers: [tex]\(5 + (-5) = 0\)[/tex].
3. Write out the quotient and remainder:
- The numbers at the bottom row become the coefficients of the quotient polynomial.
- Therefore, the quotient is written as [tex]\(2x + 5\)[/tex] (since the original polynomial was of degree 2, the result is of degree 1).
Given these calculations, the correct answer is:
C. [tex]\(2x + 5\)[/tex]
This means when you divide the polynomial [tex]\(2x^2 + 7x + 5\)[/tex] by [tex]\(x + 1\)[/tex] using synthetic division, the quotient you obtain is [tex]\(2x + 5\)[/tex].
The problem involves dividing the polynomial represented by its coefficients [tex]\([2, 7, 5]\)[/tex] by [tex]\(x + 1\)[/tex]. In synthetic division, the divisor is given in the form [tex]\(x - c\)[/tex]. Here, [tex]\(c\)[/tex] is [tex]\(-1\)[/tex] (since the expression under the division is [tex]\(x + 1\)[/tex]).
### Steps:
1. Set up the synthetic division:
- Write the coefficients of the polynomial: [tex]\(2, 7, 5\)[/tex].
- The number to the left of the division bar (the zero of the divisor) is [tex]\(-1\)[/tex].
2. Perform the synthetic division:
- Bring down the first coefficient (2) as it is.
- Multiply [tex]\(-1\)[/tex] by this number (2) and put the result under the next coefficient: [tex]\(-1 \times 2 = -2\)[/tex]. Write [tex]\(-2\)[/tex] below 7.
- Add the numbers in the second column: [tex]\(7 + (-2) = 5\)[/tex].
- Multiply [tex]\(-1\)[/tex] by the new result (5) and put this result under the next coefficient: [tex]\(-1 \times 5 = -5\)[/tex]. Write [tex]\(-5\)[/tex] below 5.
- Add these two numbers: [tex]\(5 + (-5) = 0\)[/tex].
3. Write out the quotient and remainder:
- The numbers at the bottom row become the coefficients of the quotient polynomial.
- Therefore, the quotient is written as [tex]\(2x + 5\)[/tex] (since the original polynomial was of degree 2, the result is of degree 1).
Given these calculations, the correct answer is:
C. [tex]\(2x + 5\)[/tex]
This means when you divide the polynomial [tex]\(2x^2 + 7x + 5\)[/tex] by [tex]\(x + 1\)[/tex] using synthetic division, the quotient you obtain is [tex]\(2x + 5\)[/tex].
