Question

Use long division to divide the polynomials: x3−x2+4/x+2 .(1 point

Answer 1

To divide the polynomial \( x^3 - x^2 + 4 \) by \( x + 2 \) using long division, follow these steps:

1. **Set up the long division**:
\[
\begin{array}{r|rrrr}
& 1 & -1 & 0 & 4 \\
x+2 & x^3 & -x^2 & 0 & 4 \\
\end{array}
\]

2. **Divide the leading term**: Divide the leading term \( x^3 \) by the leading term of the divisor \( x \):
\[
\frac{x^3}{x} = x^2
\]
Place \( x^2 \) above the line.

3. **Multiply**: Multiply \( x^2 \) by \( x + 2 \) and write it underneath:
\[
x^2(x + 2) = x^3 + 2x^2
\]
\[
\begin{array}{r|rrrr}
& 1 & -1 & 0 & 4 \\
x+2 & x^3 & -x^2 & 0 & 4 \\
& & x^3 & +2x^2 & \\
\hline
& & 0 & -3x^2 & 4 \\
\end{array}
\]

4. **Subtract**: Subtract the result from the previous row:
\[
0 - (-3x^2) + 0 + 4 = -3x^2 + 4
\]

5. **Repeat the process**: Now, divide the leading term \( -3x^2 \) by the leading term \( x \):
\[
\frac{-3x^2}{x} = -3x
\]
Place \( -3x \) above the line.

6. **Multiply**: Multiply \( -3x \) by \( x + 2 \):
\[
-3x(x + 2) = -3x^2 - 6x
\]
Write it beneath \(-3x^2 + 4\):
\[
\begin{array}{r|rrrr}
& 1 & -1 & 0 & 4 \\
x+2 & x^3 & -x^2 & 0 & 4 \\
& & x^3 & +2x^2 & \\
& & 0 & -3x^2 & 4 \\
& & & -3x^2 & -6x \\
\hline
& & & 0 & 10x + 4 \\
\end{array}
\]

7. **Subtract**: Subtract the result from the previous row:
\[
4 - (-6x) = 10x + 4
\]

8. **Repeat again**: Divide \( 10x \) by \( x \):
\[
\frac{10x}{x} = 10
\]
Place \( 10 \) above the line.

9. **Multiply**: Multiply \( 10 \) by \( x + 2 \):
\[
10(x + 2) = 10x + 20
\]
Write it beneath \( 10x + 4 \):
\[
\begin{array}{r|rrrr}
& 1 & -1 & 0 & 4 \\
x+2 & x^3 & -x^2 & 0 & 4 \\
& & x^3 & +2x^2 & \\
& & 0 & -3x^2 & 4 \\
& & & -3x^2 & -6x \\
& & & 0 & 10x + 4 \\
& & & & 10x + 20 \\
\hline
& & & & 0 & -16 \\
\end{array}
\]

10. **Subtract**: Subtract the results:
\[
4 - 20 = -16
\]

So the quotient is:
\[
\text{Quotient} = x^2 - 3x + 10
\]
and the remainder is:
\[
\text{Remainder} = -16
\]

Thus, the result of the polynomial long division of \( \frac{x^3 - x^2 + 4}{x + 2} \) is:
\[
x^2 - 3x + 10 - \frac{16}{x + 2}
\]