Question

Use the long division method to find the result when x, cubed, plus, 9, x, squared, plus, 21, x, plus, 9x
3
+9x
2
+21x+9 is divided by x, plus, 3x+3.

Answer 1

To use the long division method to divide the polynomial \( x^3 + 9x^2 + 21x + 9 \) by the polynomial \( x + 3 \), we can follow these steps:

### Step 1: Set up the long division

We write it in the long division format:

```
____________
x + 3 | x^3 + 9x^2 + 21x + 9
```

### Step 2: Divide

1. **First term:** Divide the first term of the dividend \( x^3 \) by the first term of the divisor \( x \):
\[
\frac{x^3}{x} = x^2
\]
Write \( x^2 \) above the division line.

2. **Multiply**: Multiply \( x^2 \) by \( x + 3 \):
\[
x^2(x + 3) = x^3 + 3x^2
\]
Write this product beneath the dividend.

3. **Subtract**: Subtract this result from the original polynomial:
\[
(x^3 + 9x^2) - (x^3 + 3x^2) = 6x^2
\]
Bring down the next term \( + 21x \):
\[
6x^2 + 21x
\]

### Step 3: Repeat the process

1. **Next term:** Divide the first term \( 6x^2 \) by \( x \):
\[
\frac{6x^2}{x} = 6x
\]
Write \( +6x \) above the division line.

2. **Multiply**: Multiply \( 6x \) by \( x + 3 \):
\[
6x(x + 3) = 6x^2 + 18x
\]
Write this product beneath \( 6x^2 + 21x \) and subtract:
\[
(6x^2 + 21x) - (6x^2 + 18x) = 3x
\]
Bring down the next term \( + 9 \):
\[
3x + 9
\]

### Step 4: One more time

1. **Next term:** Divide \( 3x \) by \( x \):
\[
\frac{3x}{x} = 3
\]
Write \( +3 \) above the division line.

2. **Multiply**: Multiply \( 3 \) by \( x + 3 \):
\[
3(x + 3) = 3x + 9
\]
Subtract:
\[
(3x + 9) - (3x + 9) = 0
\]

### Final Result

Since the remainder is zero, the final result of dividing \( x^3 + 9x^2 + 21x + 9 \) by \( x + 3 \) is:
\[
\boxed{x^2 + 6x + 3}
\]