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Answer :
To solve the synthetic division of the polynomial [tex]\(2x + 7\)[/tex] by [tex]\(x + 1\)[/tex], follow these steps:
1. Identify the dividend and the divisor:
- The polynomial [tex]\(2x + 7\)[/tex] is the dividend.
- The divisor is [tex]\(x + 1\)[/tex].
2. Use the opposite of the divisor root for synthetic division:
- The root of the divisor [tex]\(x + 1\)[/tex] is [tex]\(x = -1\)[/tex], so we use [tex]\(-1\)[/tex].
3. Set up for synthetic division:
- Write down the coefficients of the dividend polynomial. Here they are [tex]\(2\)[/tex] for [tex]\(2x\)[/tex] and [tex]\(7\)[/tex] for the constant term, so the coefficients are [2, 7].
- To the left, write the [tex]\(-1\)[/tex], since that is what you'll use for the synthetic division.
4. Perform the synthetic division:
- Bring down the first coefficient (2) directly below the line.
- Multiply: [tex]\(-1\)[/tex] by the number just brought down (2). The result is [tex]\(-2\)[/tex].
- Add: the result ([tex]\(-2\)[/tex]) to the next coefficient (7). [tex]\(7 + (-2) = 5\)[/tex].
5. Interpret the results:
- The numbers at the bottom line of your synthetic division setup show the coefficients of the quotient.
- The first number is the coefficient of [tex]\(x\)[/tex], and the second number is the remainder.
- So, the quotient polynomial is [tex]\(2x + 5\)[/tex], and the remainder is 0.
Based on the results from synthetic division, the correct option is:
C. [tex]\(2x + 5\)[/tex]
1. Identify the dividend and the divisor:
- The polynomial [tex]\(2x + 7\)[/tex] is the dividend.
- The divisor is [tex]\(x + 1\)[/tex].
2. Use the opposite of the divisor root for synthetic division:
- The root of the divisor [tex]\(x + 1\)[/tex] is [tex]\(x = -1\)[/tex], so we use [tex]\(-1\)[/tex].
3. Set up for synthetic division:
- Write down the coefficients of the dividend polynomial. Here they are [tex]\(2\)[/tex] for [tex]\(2x\)[/tex] and [tex]\(7\)[/tex] for the constant term, so the coefficients are [2, 7].
- To the left, write the [tex]\(-1\)[/tex], since that is what you'll use for the synthetic division.
4. Perform the synthetic division:
- Bring down the first coefficient (2) directly below the line.
- Multiply: [tex]\(-1\)[/tex] by the number just brought down (2). The result is [tex]\(-2\)[/tex].
- Add: the result ([tex]\(-2\)[/tex]) to the next coefficient (7). [tex]\(7 + (-2) = 5\)[/tex].
5. Interpret the results:
- The numbers at the bottom line of your synthetic division setup show the coefficients of the quotient.
- The first number is the coefficient of [tex]\(x\)[/tex], and the second number is the remainder.
- So, the quotient polynomial is [tex]\(2x + 5\)[/tex], and the remainder is 0.
Based on the results from synthetic division, the correct option is:
C. [tex]\(2x + 5\)[/tex]
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