Question

which of the following is the simplified form of (-10x^2 +20x+80)/(x+2)

Answer 1

To simplify the expression \((-10x^2 + 20x + 80)/(x + 2)\), we can use polynomial long division or synthetic division.

Let's perform polynomial long division:

1. Divide the leading term of the numerator \(-10x^2\) by the leading term of the denominator \(x\) to get \(-10x\).
2. Multiply \(-10x\) by the entire divisor \((x + 2)\):

\[
-10x(x + 2) = -10x^2 - 20x
\]

3. Subtract this result from the original numerator:

\[
(-10x^2 + 20x + 80) - (-10x^2 - 20x) = 20x + 20x + 80 = 40x + 80
\]

4. Now, divide the leading term of the new polynomial \(40x + 80\) by the leading term of the denominator \(x\):

\[
\frac{40x}{x} = 40
\]

5. Multiply the entire divisor by \(40\):

\[
40(x + 2) = 40x + 80
\]

6. Subtract this from the remaining polynomial:

\[
(40x + 80) - (40x + 80) = 0
\]

The polynomial long division gives us:

\[
-10x + 40
\]

Since there is no remainder, the simplified form of \((-10x^2 + 20x + 80)/(x + 2)\) is:

\[
\boxed{-10x + 40}
\]