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Answer :
Sure! Let's go through the synthetic division step-by-step:
We are dividing the polynomial with coefficients [tex]\(1, 5, -1, 4\)[/tex] by [tex]\(x - 2\)[/tex].
1. Set up the division:
- Write down the divisor, which is [tex]\(2\)[/tex], and the coefficients of the polynomial: [tex]\(1, 5, -1, 4\)[/tex].
2. Start the synthetic division process:
- Bring down the first coefficient [tex]\(1\)[/tex] to the bottom row. This is the starting value.
3. Multiply and add repeatedly:
- Multiply the number in the bottom row (initially [tex]\(1\)[/tex]) by the divisor ([tex]\(2\)[/tex]) and write the result below the next coefficient ([tex]\(5\)[/tex]).
- Add the number you just wrote ([tex]\(2\)[/tex]) to the next coefficient ([tex]\(5\)[/tex]). This gives [tex]\(5 + 2 = 7\)[/tex]. Write [tex]\(7\)[/tex] in the bottom row.
- Repeat this: Multiply [tex]\(7\)[/tex] by the divisor ([tex]\(2\)[/tex]), giving [tex]\(14\)[/tex]. Write it under the next coefficient ([tex]\(-1\)[/tex]).
- Add [tex]\(-1 + 14 = 13\)[/tex]. Write [tex]\(13\)[/tex] in the bottom row.
- Continue: Multiply [tex]\(13\)[/tex] by the divisor ([tex]\(2\)[/tex]), giving [tex]\(26\)[/tex]. Write it under the next coefficient ([tex]\(4\)[/tex]).
- Add [tex]\(4 + 26 = 30\)[/tex]. Write [tex]\(30\)[/tex] in the bottom row, representing the remainder.
4. Complete the division:
- The numbers in the bottom row, except the last one, represent the coefficients of the quotient. For this case, we have coefficients: [tex]\(1\)[/tex] and [tex]\(7\)[/tex].
- The remainder is [tex]\(30\)[/tex].
So the quotient in polynomial form is [tex]\(x - 7\)[/tex].
Based on this, the correct answer is:
C. [tex]\(x - 7\)[/tex]
We are dividing the polynomial with coefficients [tex]\(1, 5, -1, 4\)[/tex] by [tex]\(x - 2\)[/tex].
1. Set up the division:
- Write down the divisor, which is [tex]\(2\)[/tex], and the coefficients of the polynomial: [tex]\(1, 5, -1, 4\)[/tex].
2. Start the synthetic division process:
- Bring down the first coefficient [tex]\(1\)[/tex] to the bottom row. This is the starting value.
3. Multiply and add repeatedly:
- Multiply the number in the bottom row (initially [tex]\(1\)[/tex]) by the divisor ([tex]\(2\)[/tex]) and write the result below the next coefficient ([tex]\(5\)[/tex]).
- Add the number you just wrote ([tex]\(2\)[/tex]) to the next coefficient ([tex]\(5\)[/tex]). This gives [tex]\(5 + 2 = 7\)[/tex]. Write [tex]\(7\)[/tex] in the bottom row.
- Repeat this: Multiply [tex]\(7\)[/tex] by the divisor ([tex]\(2\)[/tex]), giving [tex]\(14\)[/tex]. Write it under the next coefficient ([tex]\(-1\)[/tex]).
- Add [tex]\(-1 + 14 = 13\)[/tex]. Write [tex]\(13\)[/tex] in the bottom row.
- Continue: Multiply [tex]\(13\)[/tex] by the divisor ([tex]\(2\)[/tex]), giving [tex]\(26\)[/tex]. Write it under the next coefficient ([tex]\(4\)[/tex]).
- Add [tex]\(4 + 26 = 30\)[/tex]. Write [tex]\(30\)[/tex] in the bottom row, representing the remainder.
4. Complete the division:
- The numbers in the bottom row, except the last one, represent the coefficients of the quotient. For this case, we have coefficients: [tex]\(1\)[/tex] and [tex]\(7\)[/tex].
- The remainder is [tex]\(30\)[/tex].
So the quotient in polynomial form is [tex]\(x - 7\)[/tex].
Based on this, the correct answer is:
C. [tex]\(x - 7\)[/tex]
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