To solve the division of the polynomial [tex]\(\frac{9x^2 + 11x - 1}{x + 2}\)[/tex], we need to perform polynomial long division. Here is a step-by-step explanation of how to do it:
1. Setup the division:
[tex]\[
\frac{9x^2 + 11x - 1}{x + 2}
\][/tex]
2. Divide the first term of the numerator by the first term of the denominator:
[tex]\[
\frac{9x^2}{x} = 9x
\][/tex]
3. Multiply the entire divisor by this result:
[tex]\[
9x \cdot (x + 2) = 9x^2 + 18x
\][/tex]
4. Subtract this product from the original polynomial:
[tex]\[
(9x^2 + 11x - 1) - (9x^2 + 18x) = (9x^2 - 9x^2) + (11x - 18x) - 1 = -7x - 1
\][/tex]
5. Repeat the process with the new polynomial:
Divide the first term of the new polynomial by the first term of the divisor:
[tex]\[
\frac{-7x}{x} = -7
\][/tex]
6. Multiply the entire divisor by this result:
[tex]\[
-7 \cdot (x + 2) = -7x - 14
\][/tex]
7. Subtract this product from the new polynomial:
[tex]\[
(-7x - 1) - (-7x - 14) = (-7x + 7x) + (-1 + 14) = 13
\][/tex]
8. State the result:
The quotient is [tex]\( 9x - 7 \)[/tex] and the remainder is [tex]\( 13 \)[/tex].
So, the final result of the division is:
[tex]\[
\frac{9x^2 + 11x - 1}{x + 2} = 9x - 7 + \frac{13}{x + 2}
\][/tex]
Thus, the quotient is [tex]\( 9x - 7 \)[/tex] and the remainder is [tex]\( 13 \)[/tex].
