Alright, let's divide the polynomial [tex]\(x^3 + x^2 + x + 2\)[/tex] by [tex]\(x^2 - 1\)[/tex] using polynomial long division. Here is the detailed, step-by-step solution:
1. Setup the division: Write the division in long division format.
[tex]\[
\frac{x^3 + x^2 + x + 2}{x^2 - 1}
\][/tex]
2. Divide the leading terms: Divide the leading term of the numerator [tex]\(x^3\)[/tex] by the leading term of the denominator [tex]\(x^2\)[/tex].
[tex]\[
\frac{x^3}{x^2} = x
\][/tex]
3. Multiply and subtract: Multiply the entire divisor [tex]\(x^2 - 1\)[/tex] by [tex]\(x\)[/tex] and subtract the result from the original polynomial:
[tex]\[
x \cdot (x^2 - 1) = x^3 - x
\][/tex]
Subtract this product from the original polynomial:
[tex]\[
(x^3 + x^2 + x + 2) - (x^3 - x) = x^2 + 2x + 2
\][/tex]
4. Repeat the division: Now, repeat the process with the new polynomial [tex]\(x^2 + 2x + 2\)[/tex].
Divide the leading term [tex]\(x^2\)[/tex] by [tex]\(x^2\)[/tex]:
[tex]\[
\frac{x^2}{x^2} = 1
\][/tex]
5. Multiply and subtract again: Multiply the entire divisor [tex]\(x^2 - 1\)[/tex] by 1 and subtract the result:
[tex]\[
1 \cdot (x^2 - 1) = x^2 - 1
\][/tex]
Subtract this product from [tex]\(x^2 + 2x + 2\)[/tex]:
[tex]\[
(x^2 + 2x + 2) - (x^2 - 1) = 2x + 3
\][/tex]
6. Conclusion: Now, the degree of the remainder [tex]\(2x + 3\)[/tex] is less than the degree of the divisor [tex]\(x^2 - 1\)[/tex], so we cannot continue the division process.
Therefore, the quotient is [tex]\(x + 1\)[/tex] and the remainder is [tex]\(2x + 3\)[/tex].
So,
[tex]\[
\frac{x^3 + x^2 + x + 2}{x^2 - 1} = x + 1 \text{ with a remainder of } 2x + 3
\][/tex]
or written in another format,
[tex]\[
x^3 + x^2 + x + 2 = (x^2 - 1)(x + 1) + (2x + 3)
\][/tex]
Thus, the quotient is [tex]\(x + 1\)[/tex] and the remainder is [tex]\(2x + 3\)[/tex].
