Answer :
To solve the division problem using synthetic division, we need to divide the polynomial [tex]\(x^3 + 5x + 3\)[/tex] by [tex]\(x + 2\)[/tex].
Here are the steps:
1. Identify the Divisor and Coefficients:
The divisor is [tex]\(x + 2\)[/tex]. We set this equal to zero to find the synthetic divisor [tex]\((x + 2 = 0 \Rightarrow x = -2)\)[/tex].
The polynomial [tex]\(x^3 + 5x + 3\)[/tex] is missing an [tex]\(x^2\)[/tex] term, so consider it as [tex]\(x^3 + 0x^2 + 5x + 3\)[/tex].
Therefore, the coefficients used in synthetic division are [tex]\(1, 0, 5, 3\)[/tex].
2. Set up the Synthetic Division:
We write down the coefficients: [tex]\(1, 0, 5, 3\)[/tex].
The number [tex]\(-2\)[/tex] is used for the synthetic division process.
3. Perform the Synthetic Division:
- Start with the first coefficient: [tex]\(1\)[/tex].
- Multiply this by [tex]\(-2\)[/tex] (the divisor) and add it to the next coefficient [tex]\(0\)[/tex], giving us:
[tex]\[
1 \times (-2) + 0 = -2
\][/tex]
- Multiply [tex]\(-2\)[/tex] by [tex]\(-2\)[/tex] and add it to the next coefficient [tex]\(5\)[/tex]:
[tex]\[
-2 \times (-2) + 5 = 9
\][/tex]
- Multiply [tex]\(9\)[/tex] by [tex]\(-2\)[/tex] and add it to the last coefficient [tex]\(3\)[/tex]:
[tex]\[
9 \times (-2) + 3 = -15
\][/tex]
4. Write the Result:
The result of synthetic division is a series of numbers: [tex]\(1, -2, 9, -15\)[/tex].
5. Interpreting the Result:
- The first numbers [tex]\(1, -2, 9\)[/tex] are the coefficients of the quotient polynomial.
- The last number [tex]\(-15\)[/tex] is the remainder.
Therefore, to fill in the blanks for the original problem:
- The first blank is [tex]\(-2\)[/tex] (the second term in the quotient).
- The second blank is [tex]\(9\)[/tex] (the third term in the quotient).
So, the blanks are filled as:
[tex]\(-2\ 9\)[/tex]
Here are the steps:
1. Identify the Divisor and Coefficients:
The divisor is [tex]\(x + 2\)[/tex]. We set this equal to zero to find the synthetic divisor [tex]\((x + 2 = 0 \Rightarrow x = -2)\)[/tex].
The polynomial [tex]\(x^3 + 5x + 3\)[/tex] is missing an [tex]\(x^2\)[/tex] term, so consider it as [tex]\(x^3 + 0x^2 + 5x + 3\)[/tex].
Therefore, the coefficients used in synthetic division are [tex]\(1, 0, 5, 3\)[/tex].
2. Set up the Synthetic Division:
We write down the coefficients: [tex]\(1, 0, 5, 3\)[/tex].
The number [tex]\(-2\)[/tex] is used for the synthetic division process.
3. Perform the Synthetic Division:
- Start with the first coefficient: [tex]\(1\)[/tex].
- Multiply this by [tex]\(-2\)[/tex] (the divisor) and add it to the next coefficient [tex]\(0\)[/tex], giving us:
[tex]\[
1 \times (-2) + 0 = -2
\][/tex]
- Multiply [tex]\(-2\)[/tex] by [tex]\(-2\)[/tex] and add it to the next coefficient [tex]\(5\)[/tex]:
[tex]\[
-2 \times (-2) + 5 = 9
\][/tex]
- Multiply [tex]\(9\)[/tex] by [tex]\(-2\)[/tex] and add it to the last coefficient [tex]\(3\)[/tex]:
[tex]\[
9 \times (-2) + 3 = -15
\][/tex]
4. Write the Result:
The result of synthetic division is a series of numbers: [tex]\(1, -2, 9, -15\)[/tex].
5. Interpreting the Result:
- The first numbers [tex]\(1, -2, 9\)[/tex] are the coefficients of the quotient polynomial.
- The last number [tex]\(-15\)[/tex] is the remainder.
Therefore, to fill in the blanks for the original problem:
- The first blank is [tex]\(-2\)[/tex] (the second term in the quotient).
- The second blank is [tex]\(9\)[/tex] (the third term in the quotient).
So, the blanks are filled as:
[tex]\(-2\ 9\)[/tex]
