Question

use long division to divide the polynomials 4x^3-21x^2+27x over x-3

Answer 1

To divide the polynomial \( 4x^3 - 21x^2 + 27x \) by \( x - 3 \) using long division, follow these steps:

1. **Set up the division**: Write \( 4x^3 - 21x^2 + 27x \) under the long division symbol and \( x - 3 \) on the outside.

```
__________________________
x - 3 | 4x^3 - 21x^2 + 27x
```

2. **Divide the first 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 the entire divisor \( x - 3 \) by \( 4x^2 \), and write the result underneath the dividend:

\[
4x^2(x - 3) = 4x^3 - 12x^2
\]

Now perform the subtraction:

\[
(4x^3 - 21x^2) - (4x^3 - 12x^2) = -21x^2 + 12x^2 = -9x^2
\]

Bring down the next term \( +27x \):

```
4x^2
__________________________
x - 3 | 4x^3 - 21x^2 + 27x
| -(4x^3 - 12x^2)
----------------------
-9x^2 + 27x
```

4. **Repeat the process**: Now divide the first term of the new polynomial \( -9x^2 \) by \( x \):

\[
\frac{-9x^2}{x} = -9x
\]

Multiply the divisor \( x - 3 \) by \( -9x \):

\[
-9x(x - 3) = -9x^2 + 27x
\]

Perform the subtraction:

\[
(-9x^2 + 27x) - (-9x^2 + 27x) = 0
\]

So we have:

```
4x^2 - 9x
__________________________
x - 3 | 4x^3 - 21x^2 + 27x
| -(4x^3 - 12x^2)
| ----------------------
-9x^2 + 27x
| -(-9x^2 + 27x)
| ----------------------
0
```

5. **Result**: Since there is no remainder, the division is complete. The quotient is:

\[
\boxed{4x^2 - 9x}
\]