Question

use the long division method to find the result when 4x^3+12x^2+17x+9 is divided by x+1

Answer 1

To perform long division on the polynomial \(4x^3 + 12x^2 + 17x + 9\) divided by \(x + 1\), follow these steps:

1. **Set up the division**: Write \(4x^3 + 12x^2 + 17x + 9\) under the long division symbol and \(x + 1\) outside.

\[
\begin{array}{r|l}
x + 1 & 4x^3 + 12x^2 + 17x + 9 \\
\end{array}
\]

2. **Divide the leading term**: Divide the leading term of the dividend \(4x^3\) by the leading term of the divisor \(x\):

\[
\frac{4x^3}{x} = 4x^2
\]

3. **Multiply and subtract**: Multiply \(4x^2\) by \(x + 1\) and subtract from the original polynomial:

\[
4x^2(x + 1) = 4x^3 + 4x^2
\]

Now subtract:

\[
(4x^3 + 12x^2 + 17x + 9) - (4x^3 + 4x^2) = (12x^2 - 4x^2) + 17x + 9 = 8x^2 + 17x + 9
\]

4. **Bring down the next term**: We now focus on \(8x^2 + 17x + 9\).

5. **Repeat the process**: Divide \(8x^2\) by \(x\):

\[
\frac{8x^2}{x} = 8x
\]

Multiply and subtract:

\[
8x(x + 1) = 8x^2 + 8x
\]

Now subtract:

\[
(8x^2 + 17x + 9) - (8x^2 + 8x) = (17x - 8x) + 9 = 9x + 9
\]

6. **Continue the division**: Now we have \(9x + 9\). Divide \(9x\) by \(x\):

\[
\frac{9x}{x} = 9
\]

Multiply and subtract:

\[
9(x + 1) = 9x + 9
\]

Now subtract:

\[
(9x + 9) - (9x + 9) = 0
\]

7. **Final result**: There are no remainder terms left, so we conclude our division. The quotient is:

\[
4x^2 + 8x + 9
\]

Thus, the result of dividing \(4x^3 + 12x^2 + 17x + 9\) by \(x + 1\) is:

\[
\boxed{4x^2 + 8x + 9}
\]