To solve the expression [tex]\(\frac{x^3 + 6x^2 - 5x}{x - 2}\)[/tex], we need to start by performing polynomial long division.
1. Set up the division:
Dividend: [tex]\(x^3 + 6x^2 - 5x\)[/tex]
Divisor: [tex]\(x - 2\)[/tex]
2. First division step:
Divide the leading term of the dividend [tex]\(x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]. This gives [tex]\(x^2\)[/tex].
Multiply [tex]\(x^2\)[/tex] by [tex]\((x - 2)\)[/tex] to get [tex]\(x^3 - 2x^2\)[/tex].
* Subtract [tex]\(x^3 - 2x^2\)[/tex] from [tex]\(x^3 + 6x^2 - 5x\)[/tex]:
[tex]\[
(x^3 + 6x^2 - 5x) - (x^3 - 2x^2) = (x^3 - x^3) + (6x^2 + 2x^2) - 5x = 8x^2 - 5x
\][/tex]
3. Second division step:
Divide the leading term of the new dividend [tex]\(8x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]. This gives [tex]\(8x\)[/tex].
Multiply [tex]\(8x\)[/tex] by [tex]\((x - 2)\)[/tex] to get [tex]\(8x^2 - 16x\)[/tex].
* Subtract [tex]\(8x^2 - 16x\)[/tex] from [tex]\(8x^2 - 5x\)[/tex]:
[tex]\[
(8x^2 - 5x) - (8x^2 - 16x) = (8x^2 - 8x^2) + (16x - 5x) = 11x
\][/tex]
4. Third division step:
Divide the leading term of the new dividend [tex]\(11x\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]. This gives [tex]\(11\)[/tex].
Multiply [tex]\(11\)[/tex] by [tex]\((x - 2)\)[/tex] to get [tex]\(11x - 22\)[/tex].
* Subtract [tex]\(11x - 22\)[/tex] from [tex]\(11x\)[/tex]:
[tex]\[
11x - (11x - 22) = 11x - 11x + 22 = 22
\][/tex]
Now we have completed the division:
[tex]\[
\frac{x^3 + 6x^2 - 5x}{x - 2} = x^2 + 8x + 11 + \frac{22}{x - 2}
\][/tex]
In the requested form, the solution is:
[tex]\[
\frac{x^3 + 6x^2 - 5x}{x - 2} = x^2 + 8x + 11 + \frac{22}{x - 2}
\][/tex]
