To find the quotient of the division problem [tex]\(\frac{x^2 + x}{x + 2}\)[/tex], we will perform polynomial long division. Here's a detailed, step-by-step solution:
1. Set up the division:
We are dividing [tex]\(x^2 + x\)[/tex] by [tex]\(x + 2\)[/tex].
2. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{x^2}{x} = x
\][/tex]
So, [tex]\(x\)[/tex] is the first term of our quotient.
3. Multiply the entire divisor by this first term of the quotient:
[tex]\[
x \cdot (x + 2) = x^2 + 2x
\][/tex]
4. Subtract this result from the original polynomial:
[tex]\[
(x^2 + x) - (x^2 + 2x) = x^2 + x - x^2 - 2x = -x
\][/tex]
5. Now, bring down any remaining term:
There are no more terms to bring down, so we proceed with the term [tex]\(-x\)[/tex].
6. Again, divide the leading term of the current polynomial by the leading term of the divisor:
[tex]\[
\frac{-x}{x} = -1
\][/tex]
So, [tex]\(-1\)[/tex] is the next term of our quotient.
7. Multiply the entire divisor by this next term of the quotient:
[tex]\[
-1 \cdot (x + 2) = -x - 2
\][/tex]
8. Subtract this result from the current polynomial:
[tex]\[
(-x) - (-x - 2) = -x + x + 2 = 2
\][/tex]
So, the division process is complete. The quotient is what we obtained in steps 2 and 6 combined:
[tex]\[
x - 1
\][/tex]
Thus, the quotient of the division problem [tex]\(\frac{x^2 + x}{x + 2}\)[/tex] is [tex]\(x - 1\)[/tex].
