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Answer :
To perform the division [tex]\((x^3 + 2x + 6) \div (x + 1)\)[/tex] using synthetic division, follow these steps:
1. Set Up Synthetic Division:
- Identify the coefficients of the polynomial [tex]\(x^3 + 2x + 6\)[/tex]. Make sure to include any missing terms with a coefficient of zero. In this case, the polynomial is [tex]\(x^3 + 0x^2 + 2x + 6\)[/tex].
- The coefficients are: [tex]\(1, 0, 2, 6\)[/tex].
2. Identify the Zero of the Divisor:
- The divisor is [tex]\(x + 1\)[/tex]. Set it equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
x + 1 = 0 \implies x = -1
\][/tex]
- Use [tex]\(-1\)[/tex] in the synthetic division process.
3. Perform Synthetic Division:
- Write down the coefficients of the dividend: [tex]\(1, 0, 2, 6\)[/tex].
- Start the synthetic division process:
- Bring down the leading coefficient [tex]\(1\)[/tex] as it is.
- Multiply this by [tex]\(-1\)[/tex] (the zero from the divisor) and add to the next coefficient:
[tex]\[
1 \times -1 + 0 = -1
\][/tex]
- Continue multiplying and adding:
[tex]\[
-1 \times -1 + 2 = 3
\][/tex]
[tex]\[
3 \times -1 + 6 = 3
\][/tex]
4. Interpreting the Result:
- The result from the synthetic division gives you the coefficients of the quotient and any remainder.
- In this case, the result is: [tex]\([1, -1, 3, 3]\)[/tex].
- The polynomial form of the quotient is [tex]\(x^2 - x + 3\)[/tex], and the remainder is [tex]\(3\)[/tex].
So, the division of [tex]\(x^3 + 2x + 6\)[/tex] by [tex]\(x + 1\)[/tex] results in a quotient of [tex]\(x^2 - x + 3\)[/tex] with a remainder of [tex]\(3\)[/tex].
1. Set Up Synthetic Division:
- Identify the coefficients of the polynomial [tex]\(x^3 + 2x + 6\)[/tex]. Make sure to include any missing terms with a coefficient of zero. In this case, the polynomial is [tex]\(x^3 + 0x^2 + 2x + 6\)[/tex].
- The coefficients are: [tex]\(1, 0, 2, 6\)[/tex].
2. Identify the Zero of the Divisor:
- The divisor is [tex]\(x + 1\)[/tex]. Set it equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
x + 1 = 0 \implies x = -1
\][/tex]
- Use [tex]\(-1\)[/tex] in the synthetic division process.
3. Perform Synthetic Division:
- Write down the coefficients of the dividend: [tex]\(1, 0, 2, 6\)[/tex].
- Start the synthetic division process:
- Bring down the leading coefficient [tex]\(1\)[/tex] as it is.
- Multiply this by [tex]\(-1\)[/tex] (the zero from the divisor) and add to the next coefficient:
[tex]\[
1 \times -1 + 0 = -1
\][/tex]
- Continue multiplying and adding:
[tex]\[
-1 \times -1 + 2 = 3
\][/tex]
[tex]\[
3 \times -1 + 6 = 3
\][/tex]
4. Interpreting the Result:
- The result from the synthetic division gives you the coefficients of the quotient and any remainder.
- In this case, the result is: [tex]\([1, -1, 3, 3]\)[/tex].
- The polynomial form of the quotient is [tex]\(x^2 - x + 3\)[/tex], and the remainder is [tex]\(3\)[/tex].
So, the division of [tex]\(x^3 + 2x + 6\)[/tex] by [tex]\(x + 1\)[/tex] results in a quotient of [tex]\(x^2 - x + 3\)[/tex] with a remainder of [tex]\(3\)[/tex].
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