To perform the polynomial division of [tex]\((x^2 + 9x - 20)\)[/tex] by [tex]\((x - 2)\)[/tex], we will follow the polynomial long division method step-by-step.
1. Set up the division:
[tex]\[
\frac{x^2 + 9x - 20}{x - 2}
\][/tex]
2. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{x^2}{x} = x
\][/tex]
This gives us the first term of the quotient: [tex]\(x\)[/tex].
3. Multiply the entire divisor by this term [tex]\(x\)[/tex] and subtract from the dividend:
[tex]\[
(x - 2) \cdot x = x^2 - 2x
\][/tex]
[tex]\[
(x^2 + 9x - 20) - (x^2 - 2x) = 11x - 20
\][/tex]
4. Repeat the process with the new polynomial [tex]\(11x - 20\)[/tex]:
[tex]\[
\frac{11x}{x} = 11
\][/tex]
This gives us the next term of the quotient: [tex]\(11\)[/tex].
5. Multiply the entire divisor by this term [tex]\(11\)[/tex] and subtract from the new polynomial:
[tex]\[
(x - 2) \cdot 11 = 11x - 22
\][/tex]
[tex]\[
(11x - 20) - (11x - 22) = 2
\][/tex]
6. Conclusion: Since [tex]\(2\)[/tex] is of lower degree than the divisor [tex]\((x - 2)\)[/tex], this is the remainder.
Thus, the quotient is [tex]\(x + 11\)[/tex] and the remainder is [tex]\(2\)[/tex]. So we can write:
[tex]\[
\frac{x^2 + 9x - 20}{x - 2} = x + 11 + \frac{2}{x - 2}
\][/tex]
Therefore, the division [tex]\( (x^2 + 9x - 20) \div (x - 2) \)[/tex] results in:
[tex]\[
x + 11 + \frac{2}{x-2}
\][/tex]
Hence, the complete polynomial division gives:
[tex]\[
x + 11 + \frac{2}{x-2}
\][/tex]
