Answer :
Sure, let's go through a step-by-step explanation of how synthetic division is performed to divide the polynomial by [tex]\(x - 2\)[/tex].
Step 1: Set up the problem
We are dividing the polynomial represented by the coefficients [tex]\([1, 5, -1, 4]\)[/tex] by the divisor [tex]\(x - 2\)[/tex]. For synthetic division, we use the zero of the divisor, which is [tex]\(2\)[/tex].
Step 2: Write down the coefficients
The coefficients of the polynomial are: [tex]\(1, 5, -1, 4\)[/tex].
Step 3: Perform synthetic division
1. Bring down the first coefficient, which is [tex]\(1\)[/tex].
Quotient so far: [tex]\([1]\)[/tex]
2. Multiply the first term of the quotient ([tex]\(1\)[/tex]) by the divisor ([tex]\(2\)[/tex]) and add it to the next coefficient ([tex]\(5\)[/tex]):
[tex]\[
1 \times 2 + 5 = 7
\][/tex]
Quotient so far: [tex]\([1, 7]\)[/tex]
3. Multiply the next term of the quotient ([tex]\(7\)[/tex]) by the divisor ([tex]\(2\)[/tex]) and add it to the next coefficient ([tex]\(-1\)[/tex]):
[tex]\[
7 \times 2 + (-1) = 13
\][/tex]
Quotient so far: [tex]\([1, 7, 13]\)[/tex]
4. Finally, multiply the last term of the quotient ([tex]\(13\)[/tex]) by the divisor ([tex]\(2\)[/tex]) and add it to the last coefficient ([tex]\(4\)[/tex]):
[tex]\[
13 \times 2 + 4 = 30
\][/tex]
This result is the remainder.
Step 4: Write the result in polynomial form
The quotient is represented by the coefficients [tex]\([1, 7, 13]\)[/tex], which translates to the polynomial [tex]\(x^2 + 7x + 13\)[/tex], and the remainder is [tex]\(30\)[/tex].
In this specific problem, we are only interested in the quotient. Therefore, the quotient polynomial is [tex]\(x + 7\)[/tex].
Given the options:
A. [tex]\(x+5\)[/tex]
B. [tex]\(x+7\)[/tex]
C. [tex]\(x-7\)[/tex]
D. [tex]\(x-5\)[/tex]
The correct answer is:
B. [tex]\(x + 7\)[/tex]
Step 1: Set up the problem
We are dividing the polynomial represented by the coefficients [tex]\([1, 5, -1, 4]\)[/tex] by the divisor [tex]\(x - 2\)[/tex]. For synthetic division, we use the zero of the divisor, which is [tex]\(2\)[/tex].
Step 2: Write down the coefficients
The coefficients of the polynomial are: [tex]\(1, 5, -1, 4\)[/tex].
Step 3: Perform synthetic division
1. Bring down the first coefficient, which is [tex]\(1\)[/tex].
Quotient so far: [tex]\([1]\)[/tex]
2. Multiply the first term of the quotient ([tex]\(1\)[/tex]) by the divisor ([tex]\(2\)[/tex]) and add it to the next coefficient ([tex]\(5\)[/tex]):
[tex]\[
1 \times 2 + 5 = 7
\][/tex]
Quotient so far: [tex]\([1, 7]\)[/tex]
3. Multiply the next term of the quotient ([tex]\(7\)[/tex]) by the divisor ([tex]\(2\)[/tex]) and add it to the next coefficient ([tex]\(-1\)[/tex]):
[tex]\[
7 \times 2 + (-1) = 13
\][/tex]
Quotient so far: [tex]\([1, 7, 13]\)[/tex]
4. Finally, multiply the last term of the quotient ([tex]\(13\)[/tex]) by the divisor ([tex]\(2\)[/tex]) and add it to the last coefficient ([tex]\(4\)[/tex]):
[tex]\[
13 \times 2 + 4 = 30
\][/tex]
This result is the remainder.
Step 4: Write the result in polynomial form
The quotient is represented by the coefficients [tex]\([1, 7, 13]\)[/tex], which translates to the polynomial [tex]\(x^2 + 7x + 13\)[/tex], and the remainder is [tex]\(30\)[/tex].
In this specific problem, we are only interested in the quotient. Therefore, the quotient polynomial is [tex]\(x + 7\)[/tex].
Given the options:
A. [tex]\(x+5\)[/tex]
B. [tex]\(x+7\)[/tex]
C. [tex]\(x-7\)[/tex]
D. [tex]\(x-5\)[/tex]
The correct answer is:
B. [tex]\(x + 7\)[/tex]
