Answer :
To solve the synthetic division problem, we are dividing a polynomial by [tex]\( x - 2 \)[/tex], which means our divider is 2. The polynomial given has coefficients: [tex]\( 1 \)[/tex], [tex]\( 5 \)[/tex], [tex]\( -1 \)[/tex], and [tex]\( 4 \)[/tex].
Here's the step-by-step process for synthetic division:
1. Setup: Write down the coefficients of the polynomial: [tex]\( 1, 5, -1, 4 \)[/tex].
2. Divider: The divisor is [tex]\( x - 2 \)[/tex], so the divider is [tex]\( 2 \)[/tex].
3. Initial Step: Bring down the first coefficient (1) directly as it is.
4. Calculation Steps:
- Multiply the current result (1) by the divider (2). This gives [tex]\( 1 \times 2 = 2 \)[/tex].
- Add this result to the next coefficient (5): [tex]\( 5 + 2 = 7 \)[/tex]. Write 7 below the line.
- Now, multiply 7 by the divider (2): [tex]\( 7 \times 2 = 14 \)[/tex].
- Add this result to the next coefficient (-1): [tex]\( -1 + 14 = 13 \)[/tex]. Write 13 below the line.
- Multiply 13 by the divider (2): [tex]\( 13 \times 2 = 26 \)[/tex].
- Add this result to the last coefficient (4): [tex]\( 4 + 26 = 30 \)[/tex]. Write 30 below the line.
5. Results: The numbers below the line, from left to right, are [tex]\( 1, 7, 13, 30 \)[/tex].
6. Quotient and Remainder:
- The first three numbers (1, 7, 13) are the coefficients of the quotient polynomial.
- Since we started with a cubic polynomial, the quotient will be one degree lower, which is quadratic.
- The quotient polynomial is: [tex]\( x^2 + 7x + 13 \)[/tex].
- The last number, 30, is the remainder.
Given the results:
[tex]\[ \text{Quotient: } x + 7 \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{x + 7} \][/tex]
Here's the step-by-step process for synthetic division:
1. Setup: Write down the coefficients of the polynomial: [tex]\( 1, 5, -1, 4 \)[/tex].
2. Divider: The divisor is [tex]\( x - 2 \)[/tex], so the divider is [tex]\( 2 \)[/tex].
3. Initial Step: Bring down the first coefficient (1) directly as it is.
4. Calculation Steps:
- Multiply the current result (1) by the divider (2). This gives [tex]\( 1 \times 2 = 2 \)[/tex].
- Add this result to the next coefficient (5): [tex]\( 5 + 2 = 7 \)[/tex]. Write 7 below the line.
- Now, multiply 7 by the divider (2): [tex]\( 7 \times 2 = 14 \)[/tex].
- Add this result to the next coefficient (-1): [tex]\( -1 + 14 = 13 \)[/tex]. Write 13 below the line.
- Multiply 13 by the divider (2): [tex]\( 13 \times 2 = 26 \)[/tex].
- Add this result to the last coefficient (4): [tex]\( 4 + 26 = 30 \)[/tex]. Write 30 below the line.
5. Results: The numbers below the line, from left to right, are [tex]\( 1, 7, 13, 30 \)[/tex].
6. Quotient and Remainder:
- The first three numbers (1, 7, 13) are the coefficients of the quotient polynomial.
- Since we started with a cubic polynomial, the quotient will be one degree lower, which is quadratic.
- The quotient polynomial is: [tex]\( x^2 + 7x + 13 \)[/tex].
- The last number, 30, is the remainder.
Given the results:
[tex]\[ \text{Quotient: } x + 7 \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{x + 7} \][/tex]
