https://file.aiquickdraw.com/mathbot/file/1728950754475wwdgeuae.jpg

The given function is:

\[
f(x) = x^4 + 5x^3 - 25x^2 - 125x
\]

We are asked to factor this polynomial, and we are given a graph to help with that.

### Step 1: Factor out the greatest common factor (GCF).
The GCF of all the terms is \( x \), so we factor that out first:

\[
f(x) = x(x^3 + 5x^2 - 25x - 125)
\]

### Step 2: Factor the cubic polynomial.
Now, we focus on factoring the cubic part, \( x^3 + 5x^2 - 25x - 125 \).

We can use **synthetic division** or observe the graph to find possible roots. From the graph, we can estimate that \( x = -5 \) is a root.

### Step 3: Use synthetic division to divide the cubic by \( (x + 5) \).

Performing synthetic division on \( x^3 + 5x^2 - 25x - 125 \) by \( x + 5 \):

\[
\begin{array}{r|rrrr}
-5 & 1 & 5 & -25 & -125 \\
   &   & -5 & 0   & 125  \\
\hline
   & 1 & 0  & -25 & 0   \\
\end{array}
\]

This gives a quotient of \( x^2 - 25 \), so we now have:

\[
f(x) = x(x + 5)(x^2 - 25)
\]

### Step 4: Factor \( x^2 - 25 \).
The expression \( x^2 - 25 \) is a difference of squares, so it factors as:

\[
x^2 - 25 = (x - 5)(x + 5)
\]

### Final Factored Form:
The completely factored form of \( f(x) \) is:

\[
f(x) = x(x + 5)^2(x - 5)
\]

### Conclusion:
The factored form is:

\[
f(x) = x(x + 5)^2(x - 5)
\]

---

Would you like further clarification on any of the steps?

Here are 5 related questions:
1. How do you perform synthetic division in more detail?
2. How do you identify possible roots from the graph of a polynomial?
3. Can all polynomials be factored using the difference of squares technique?
4. What is the role of the greatest common factor in factoring?
5. How do you factor higher-degree polynomials with complex roots?

**Tip:** Always check the graph for hints about possible real roots when factoring polynomials. It can simplify the process significantly!

The function f(x) = x^4 + 5x^3 - 25x^2 - 125x has the graph as shown. What is the factored form of the polynomial?

Learn how to factor the polynomial f(x) = x^4 + 5x^3 - 25x^2 - 125x using the greatest common factor (GCF), synthetic division, and the difference of squares technique. Suitable for students in Grades 10-12.

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