Use synthetic division to divide the polynomials. $$ \frac{x^{4}-625}{x+5}=\square $$

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To perform synthetic division of the polynomial $$ x^4 - 625 $$ by $$ x + 5 $$, we first need to set up the synthetic division. The divisor $$ x + 5 $$ corresponds to the root $$ -5 $$.

### Step 1: Set up the synthetic division

1. Write down the coefficients of the polynomial $$ x^4 - 625 $$. Since this is a fourth-degree polynomial, we need to include coefficients for all powers of $$ x $$:
   - Coefficient of $$ x^4 $$: 1
   - Coefficient of $$ x^3 $$: 0 (since there is no $$ x^3 $$ term)
   - Coefficient of $$ x^2 $$: 0 (since there is no $$ x^2 $$ term)
   - Coefficient of $$ x^1 $$: 0 (since there is no $$ x^1 $$ term)
   - Coefficient of $$ x^0 $$: -625

Thus, the coefficients are: $$ [1, 0, 0, 0, -625] $$.

2. Write $$ -5 $$ to the left of the coefficients.

### Step 2: Perform synthetic division

Now we will perform the synthetic division:

```
-5 |  1   0   0   0   -625
   |      -5   25  -125  625
   -------------------------
     1  -5   25  -125   0
```

### Step 3: Interpret the result

The bottom row gives us the coefficients of the quotient polynomial and the remainder:

- The coefficients $$ 1, -5, 25, -125 $$ correspond to the polynomial $$ x^3 - 5x^2 + 25x - 125 $$.
- The remainder is $$ 0 $$.

### Final Result

Thus, the result of the division is:

$$
\frac{x^4 - 625}{x + 5} = x^3 - 5x^2 + 25x - 125
$$

So, the answer is:

$$
\boxed{x^3 - 5x^2 + 25x - 125}
$$
The result of dividing $$ x^4 - 625 $$ by $$ x + 5 $$ is $$ x^3 - 5x^2 + 25x - 125 $$.

Use synthetic division to divide the polynomials. \( \frac{x^{4}-625}{x+5}=\square \)

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