Answer :
Sure! Let's solve the synthetic division problem step-by-step to find the quotient in polynomial form.
We are dividing the polynomial represented by the coefficients [tex]\(1\)[/tex], [tex]\(5\)[/tex], [tex]\(-1\)[/tex], and [tex]\(4\)[/tex] by [tex]\(x - 2\)[/tex].
Here's how we perform synthetic division:
1. Write down the coefficients of the polynomial: 1, 5, -1, and 4.
2. Set up the division with 2 (from the divisor [tex]\(x - 2\)[/tex]) to the left. We'll work through the coefficients one by one.
3. Bring down the leading coefficient (1) since it starts the division process. This becomes the first coefficient of the quotient.
4. Multiply this leading coefficient (1) by 2 (the number on the left), which gives 2. Then, add this result to the next coefficient in the list (5):
[tex]\[
2 + 5 = 7
\][/tex]
Write 7 below the line as the second coefficient of the quotient.
5. Multiply 7 (the result we just found) by 2:
[tex]\[
7 \times 2 = 14
\][/tex]
Add this 14 to the next coefficient (-1):
[tex]\[
14 + (-1) = 13
\][/tex]
Write 13 below the line as the third coefficient of the quotient.
6. Multiply 13 by 2:
[tex]\[
13 \times 2 = 26
\][/tex]
Add this 26 to the final coefficient (4):
[tex]\[
26 + 4 = 30
\][/tex]
This result is the remainder.
7. With the synthetic division complete, the quotient is given by the coefficients we found: 1 and 7, plus a remainder of 30.
Therefore, the quotient polynomial is [tex]\(x + 7\)[/tex], which corresponds to option B.
The remainder does not affect the quotient in polynomial form, as we are asked only for the quotient itself.
We are dividing the polynomial represented by the coefficients [tex]\(1\)[/tex], [tex]\(5\)[/tex], [tex]\(-1\)[/tex], and [tex]\(4\)[/tex] by [tex]\(x - 2\)[/tex].
Here's how we perform synthetic division:
1. Write down the coefficients of the polynomial: 1, 5, -1, and 4.
2. Set up the division with 2 (from the divisor [tex]\(x - 2\)[/tex]) to the left. We'll work through the coefficients one by one.
3. Bring down the leading coefficient (1) since it starts the division process. This becomes the first coefficient of the quotient.
4. Multiply this leading coefficient (1) by 2 (the number on the left), which gives 2. Then, add this result to the next coefficient in the list (5):
[tex]\[
2 + 5 = 7
\][/tex]
Write 7 below the line as the second coefficient of the quotient.
5. Multiply 7 (the result we just found) by 2:
[tex]\[
7 \times 2 = 14
\][/tex]
Add this 14 to the next coefficient (-1):
[tex]\[
14 + (-1) = 13
\][/tex]
Write 13 below the line as the third coefficient of the quotient.
6. Multiply 13 by 2:
[tex]\[
13 \times 2 = 26
\][/tex]
Add this 26 to the final coefficient (4):
[tex]\[
26 + 4 = 30
\][/tex]
This result is the remainder.
7. With the synthetic division complete, the quotient is given by the coefficients we found: 1 and 7, plus a remainder of 30.
Therefore, the quotient polynomial is [tex]\(x + 7\)[/tex], which corresponds to option B.
The remainder does not affect the quotient in polynomial form, as we are asked only for the quotient itself.
