Certainly! Let's perform the long division step-by-step for dividing [tex]\( x^3 + x^2 + x + 2 \)[/tex] by [tex]\( x^2 - 1 \)[/tex].
We will write this in the form of long division, where the dividend is [tex]\( x^3 + x^2 + x + 2 \)[/tex] and the divisor is [tex]\( x^2 - 1 \)[/tex].
1. Divide the leading terms:
- The leading term of the dividend is [tex]\( x^3 \)[/tex].
- The leading term of the divisor is [tex]\( x^2 \)[/tex].
Divide [tex]\( x^3 \)[/tex] by [tex]\( x^2 \)[/tex]:
[tex]\[
\frac{x^3}{x^2} = x
\][/tex]
2. Multiply and subtract:
- Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^2 - 1 \)[/tex]:
[tex]\[
x \cdot (x^2 - 1) = x^3 - x
\][/tex]
- Subtract this from the original polynomial [tex]\( x^3 + x^2 + x + 2 \)[/tex]:
[tex]\[
(x^3 + x^2 + x + 2) - (x^3 - x) = x^2 + 2x + 2
\][/tex]
3. Repeat the process:
- Now the new dividend is [tex]\( x^2 + 2x + 2 \)[/tex].
Divide the leading term [tex]\( x^2 \)[/tex] by the leading term of the divisor [tex]\( x^2 \)[/tex]:
[tex]\[
\frac{x^2}{x^2} = 1
\][/tex]
- Multiply [tex]\( 1 \)[/tex] by the divisor [tex]\( x^2 - 1 \)[/tex]:
[tex]\[
1 \cdot (x^2 - 1) = x^2 - 1
\][/tex]
- Subtract this result from the current dividend:
[tex]\[
(x^2 + 2x + 2) - (x^2 - 1) = 2x + 3
\][/tex]
4. Conclusion:
- The quotient obtained by dividing [tex]\( x^3 + x^2 + x + 2 \)[/tex] by [tex]\( x^2 - 1 \)[/tex] is [tex]\( x + 1 \)[/tex].
- The remainder after the division is [tex]\( 2x + 3 \)[/tex].
So, the result of the division of [tex]\( x^3 + x^2 + x + 2 \)[/tex] by [tex]\( x^2 - 1 \)[/tex] can be expressed as:
[tex]\[
\text{Quotient: } x + 1
\][/tex]
[tex]\[
\text{Remainder: } 2x + 3
\][/tex]
Thus, when [tex]\( x^3 + x^2 + x + 2 \)[/tex] is divided by [tex]\( x^2 - 1 \)[/tex], we get:
[tex]\[
\boxed{(x + 1, \, 2x + 3)}
\][/tex]
