Answer :
Sure! Let's go through the synthetic division process step-by-step.
We are given the polynomial [tex]\(2x^2 + 7x + 5\)[/tex] and we need to divide it by [tex]\(x + 1\)[/tex] using synthetic division.
### Steps for Synthetic Division:
1. Set up the divisor: Since we are dividing by [tex]\(x + 1\)[/tex], we take the opposite sign of the constant term from the divisor, which is [tex]\(-1\)[/tex].
2. Write down the coefficients: We take the coefficients of the polynomial [tex]\(2x^2 + 7x + 5\)[/tex], which are [tex]\(2\)[/tex], [tex]\(7\)[/tex], and [tex]\(5\)[/tex].
3. Bring down the leading coefficient: Start with the leading coefficient, which is [tex]\(2\)[/tex].
4. Multiply and add:
- Multiply [tex]\(-1\)[/tex] (the value from the divisor) with [tex]\(2\)[/tex] (the current value) and write the result under the next coefficient.
- The result is [tex]\(-1 \times 2 = -2\)[/tex].
- Add this result to the next coefficient: [tex]\(7 + (-2) = 5\)[/tex].
5. Repeat the multiply and add process:
- Multiply [tex]\(-1\)[/tex] by the new sum, [tex]\(5\)[/tex]: [tex]\(-1 \times 5 = -5\)[/tex].
- Add this result to the next coefficient: [tex]\(5 + (-5) = 0\)[/tex].
6. Interpret the result:
- The numbers in the bottom row, excluding the last number (remainder), are the coefficients of the quotient.
- In this case, they are [tex]\(2\)[/tex] and [tex]\(5\)[/tex].
The quotient is therefore [tex]\(2x + 5\)[/tex].
### Final Answer
The quotient of the division of [tex]\(2x^2 + 7x + 5\)[/tex] by [tex]\(x + 1\)[/tex] is [tex]\(2x + 5\)[/tex].
Let me know if you have any questions or need further clarification!
We are given the polynomial [tex]\(2x^2 + 7x + 5\)[/tex] and we need to divide it by [tex]\(x + 1\)[/tex] using synthetic division.
### Steps for Synthetic Division:
1. Set up the divisor: Since we are dividing by [tex]\(x + 1\)[/tex], we take the opposite sign of the constant term from the divisor, which is [tex]\(-1\)[/tex].
2. Write down the coefficients: We take the coefficients of the polynomial [tex]\(2x^2 + 7x + 5\)[/tex], which are [tex]\(2\)[/tex], [tex]\(7\)[/tex], and [tex]\(5\)[/tex].
3. Bring down the leading coefficient: Start with the leading coefficient, which is [tex]\(2\)[/tex].
4. Multiply and add:
- Multiply [tex]\(-1\)[/tex] (the value from the divisor) with [tex]\(2\)[/tex] (the current value) and write the result under the next coefficient.
- The result is [tex]\(-1 \times 2 = -2\)[/tex].
- Add this result to the next coefficient: [tex]\(7 + (-2) = 5\)[/tex].
5. Repeat the multiply and add process:
- Multiply [tex]\(-1\)[/tex] by the new sum, [tex]\(5\)[/tex]: [tex]\(-1 \times 5 = -5\)[/tex].
- Add this result to the next coefficient: [tex]\(5 + (-5) = 0\)[/tex].
6. Interpret the result:
- The numbers in the bottom row, excluding the last number (remainder), are the coefficients of the quotient.
- In this case, they are [tex]\(2\)[/tex] and [tex]\(5\)[/tex].
The quotient is therefore [tex]\(2x + 5\)[/tex].
### Final Answer
The quotient of the division of [tex]\(2x^2 + 7x + 5\)[/tex] by [tex]\(x + 1\)[/tex] is [tex]\(2x + 5\)[/tex].
Let me know if you have any questions or need further clarification!
