Let's solve the division of the polynomial [tex]\(\frac{x^4 - x^3 + x^2 - x + 2}{x - 2}\)[/tex] using polynomial long division.
1. Set up the division: Divide [tex]\(x^4 - x^3 + x^2 - x + 2\)[/tex] by [tex]\(x - 2\)[/tex].
[tex]\[
\begin{array}{r|rrrrr}
x^3 + x^2 + 3x + 5 & x - 2 & \text{remainder}\\
\hline
x^4 - x^3 + x^2 - x + 2 & &
\end{array}
\][/tex]
2. First term: Divide the leading term [tex]\(x^4\)[/tex] by the leading term [tex]\(x\)[/tex] of the divisor.
[tex]\[ x^4 \div x = x^3 \][/tex]
3. Multiply and subtract:
- Multiply [tex]\(x^3\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ x^3 \cdot (x - 2) = x^4 - 2x^3 \][/tex]
- Subtract this product from [tex]\(x^4 - x^3 + x^2 - x + 2\)[/tex]:
[tex]\[
(x^4 - x^3 + x^2 - x + 2) - (x^4 - 2x^3) = x^4 - x^3 + x^2 - x + 2 - x^4 + 2x^3 = x^3 + x^2 - x + 2
\][/tex]
4. Second term: Divide the new leading term [tex]\(x^3\)[/tex] by the leading term [tex]\(x\)[/tex] of the divisor.
[tex]\[ x^3 \div x = x^2 \][/tex]
5. Multiply and subtract:
- Multiply [tex]\(x^2\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ x^2 \cdot (x - 2) = x^3 - 2x^2 \][/tex]
- Subtract this product from [tex]\(x^3 + x^2 - x + 2\)[/tex]:
[tex]\[
(x^3 + x^2 - x + 2) - (x^3 - 2x^2) = x^3 + x^2 - x + 2 - x^3 + 2x^2 = 3x^2 - x + 2
\][/tex]
6. Third term: Divide the new leading term [tex]\(3x^2\)[/tex] by the leading term [tex]\(x\)[/tex] of the divisor.
[tex]\[ 3x^2 \div x = 3x \][/tex]
7. Multiply and subtract:
- Multiply [tex]\(3x\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ 3x \cdot (x - 2) = 3x^2 - 6x \][/tex]
- Subtract this product from [tex]\(3x^2 - x + 2\)[/tex]:
[tex]\[
(3x^2 - x + 2) - (3x^2 - 6x) = 3x^2 - x + 2 - 3x^2 + 6x = 5x + 2
\][/tex]
8. Fourth term: Divide the new leading term [tex]\(5x\)[/tex] by the leading term [tex]\(x\)[/tex] of the divisor.
[tex]\[ 5x \div x = 5 \][/tex]
9. Multiply and subtract:
- Multiply [tex]\(5\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ 5 \cdot (x - 2) = 5x - 10 \][/tex]
- Subtract this product from [tex]\(5x + 2\)[/tex]:
[tex]\[
(5x + 2) - (5x - 10) = 5x + 2 - 5x + 10 = 12
\][/tex]
10. Result: The quotient is [tex]\(x^3 + x^2 + 3x + 5\)[/tex] and the remainder is 12. Therefore, the division of [tex]\(\frac{x^4 - x^3 + x^2 - x + 2}{x - 2}\)[/tex] results in:
[tex]\[
\frac{x^4 - x^3 + x^2 - x + 2}{x - 2} = x^3 + x^2 + 3x + 5 + \frac{12}{x - 2}
\][/tex]
