To solve the polynomial division [tex]\( \frac{x^5 - 4x^3 + x + 1}{x^2 - 1} \)[/tex], we'll use long division of polynomials. Let's go through this step-by-step:
1. Set up the division:
[tex]\[
\frac{x^5 - 4x^3 + x + 1}{x^2 - 1}
\][/tex]
2. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[
\frac{x^5}{x^2} = x^3
\][/tex]
So, our first term in the quotient is [tex]\( x^3 \)[/tex].
3. Multiply the entire denominator by [tex]\( x^3 \)[/tex]:
[tex]\[
x^3 \cdot (x^2 - 1) = x^5 - x^3
\][/tex]
4. Subtract this result from the original polynomial:
[tex]\[
(x^5 - 4x^3 + x + 1) - (x^5 - x^3) = -3x^3 + x + 1
\][/tex]
Now, our new polynomial is [tex]\( -3x^3 + x + 1 \)[/tex].
5. Repeat the process with the new polynomial. Divide the new leading term by the leading term of the denominator:
[tex]\[
\frac{-3x^3}{x^2} = -3x
\][/tex]
So, our next term in the quotient is [tex]\( -3x \)[/tex].
6. Multiply the entire denominator by [tex]\( -3x \)[/tex]:
[tex]\[
-3x \cdot (x^2 - 1) = -3x^3 + 3x
\][/tex]
7. Subtract this result from the new polynomial:
[tex]\[
(-3x^3 + x + 1) - (-3x^3 + 3x) = -2x + 1
\][/tex]
Now, our new polynomial is [tex]\( -2x + 1 \)[/tex].
8. Divide the leading term of the new polynomial by the leading term of the denominator:
[tex]\[
\frac{-2x}{x^2} = -2 \cdot \frac{1}{x}
\][/tex]
However, since [tex]\(\frac{-2x}{x^2}\)[/tex] does not provide a polynomial term without a negative power of [tex]\(x\)[/tex], our polynomial division stops here.
9. The quotient and the remainder:
- Quotient: By combining the terms we have calculated, the quotient is [tex]\( x^3 - 3x \)[/tex].
- Remainder: The remaining part of the polynomial that we could not divide further is [tex]\( -2x + 1 \)[/tex].
Thus, the polynomial division gives us the final result:
[tex]\[
\frac{x^5 - 4x^3 + x + 1}{x^2 - 1} = x^3 - 3x \quad \text{with a remainder of} \quad -2x + 1
\][/tex]
So we can write:
[tex]\[
x^5 - 4x^3 + x + 1 = (x^2 - 1)(x^3 - 3x) + (-2x + 1)
\][/tex]
