Sure, let's go through the process of polynomial long division step-by-step to divide [tex]\( (x^3 + x^2 + x + 2) \)[/tex] by [tex]\( (x^2 - 1) \)[/tex].
1. Setup the Division:
- Dividend: [tex]\( x^3 + x^2 + x + 2 \)[/tex]
- Divisor: [tex]\( x^2 - 1 \)[/tex]
2. First Division Step:
- Divide the leading term of the dividend [tex]\( x^3 \)[/tex] by the leading term of the divisor [tex]\( x^2 \)[/tex] to get [tex]\( x \)[/tex].
- Multiply the entire divisor [tex]\( x^2 - 1 \)[/tex] by [tex]\( x \)[/tex]:
[tex]\( x \cdot (x^2 - 1) = x^3 - x \)[/tex].
- Subtract this from the current dividend [tex]\( x^3 + x^2 + x + 2 \)[/tex]:
[tex]\((x^3 + x^2 + x + 2) - (x^3 - x) = x^2 + 2x + 2\)[/tex].
3. Second Division Step:
- Now repeat the process with the new dividend [tex]\( x^2 + 2x + 2 \)[/tex].
- Divide the leading term of the new dividend [tex]\( x^2 \)[/tex] by the leading term of the divisor [tex]\( x^2 \)[/tex] to get [tex]\( 1 \)[/tex].
- Multiply the entire divisor [tex]\( x^2 - 1 \)[/tex] by [tex]\( 1 \)[/tex]:
[tex]\( 1 \cdot (x^2 - 1) = x^2 - 1 \)[/tex].
- Subtract this from the current dividend [tex]\( x^2 + 2x + 2 \)[/tex]:
[tex]\((x^2 + 2x + 2) - (x^2 - 1) = 2x + 3\)[/tex].
4. Conclusion:
- The division quotient is [tex]\( x + 1 \)[/tex].
- The remainder is [tex]\( 2x + 3 \)[/tex].
Putting it all together, we have:
[tex]\[
\frac{x^3 + x^2 + x + 2}{x^2 - 1} = x + 1 \, \text{with a remainder of} \, 2x + 3.
\][/tex]
Therefore, the division result can be expressed as:
[tex]\[
x + 1 \quad \text{(quotient)} \quad \text{and} \quad 2x + 3 \quad \text{(remainder)}.
\][/tex]
So, in full, the result is:
[tex]\[
1.0, 1.0 \quad \text{(quotient coefficients)}
\][/tex]
[tex]\[
2.0, 3.0 \quad \text{(remainder coefficients)}
\][/tex]
