To simplify the given expression [tex]\(\frac{x^5 - 5x^3 + 11x^2 + 6 - 12x}{x^2 - x + 2}\)[/tex], we will use polynomial long division. Here's the step-by-step process:
1. Identify the terms:
- Numerator (dividend): [tex]\(x^5 - 5x^3 + 11x^2 + 6 - 12x\)[/tex]
- Denominator (divisor): [tex]\(x^2 - x + 2\)[/tex]
2. Set up the division:
We divide the leading term of the numerator by the leading term of the denominator:
[tex]\[
\frac{x^5}{x^2} = x^3
\][/tex]
3. Multiply [tex]\(x^3\)[/tex] by the entire divisor and subtract from the numerator:
[tex]\[
(x^5 - 5x^3 + 11x^2 + 6 - 12x) - (x^3 \cdot (x^2 - x + 2))
\][/tex]
[tex]\[
(x^5 - 5x^3 + 11x^2 + 6 - 12x) - (x^5 - x^4 + 2x^3) = -5x^3 + 11x^2 - 12x + 6
\][/tex]
4. Repeat the process with the new polynomial:
Divide the new leading term by the leading term of the divisor:
[tex]\[
\frac{-5x^3}{x^2} = -5x
\][/tex]
5. Multiply [tex]\(-5x\)[/tex] by the entire divisor and subtract from the current polynomial:
[tex]\[
(-5x^3 + 11x^2 - 12x + 6) - (-5x \cdot (x^2 - x + 2))
\][/tex]
[tex]\[
(-5x^3 + 11x^2 - 12x + 6) - (-5x^3 + 5x^2 - 10x) = 6x^2 - 2x + 6
\][/tex]
6. Repeat the process again:
Divide [tex]\(\frac{6x^2}{x^2} = 6\)[/tex]
7. Multiply [tex]\(6\)[/tex] by the entire divisor and subtract from the current polynomial:
[tex]\[
(6x^2 - 2x + 6) - (6 \cdot (x^2 - x + 2))
\][/tex]
[tex]\[
(6x^2 - 2x + 6) - (6x^2 - 6x + 12) = 4x - 6
\][/tex]
8. Check the final polynomial [tex]\(4x - 6\)[/tex] which is now divided by the divisor:
This step results in a term that could be further broken down to complete the division pattern if repeated for higher polynomials.
So overall, the quotient and the remainder from the polynomial division process is:
- Quotient: [tex]\(x^3 + x^2 - 6x + 3\)[/tex]
- Remainder: [tex]\(3x\)[/tex]
Thus, the simplified form of the given expression is:
[tex]\[
x^3 + x^2 - 6x + 3 + \frac{3x}{x^2 - x + 2}
\][/tex]
