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Let's solve the polynomial \( -25x + 50 + x^3 - 2x^2 = 0 \) and express it in factored form as requested in the image.

### Step 1: Rewrite in Standard Form
The given polynomial is already close to standard form but we can rearrange it as:
\[
x^3 - 2x^2 - 25x + 50 = 0
\]
This is the standard form of the cubic equation.

### Step 2: Use Synthetic Division to Find Roots
To use synthetic division, we first try to find possible rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (50) divided by the factors of the leading coefficient (1). So, the possible roots are:
\[
\pm 1, \pm 2, \pm 5, \pm 10, \pm 25, \pm 50
\]

Let's try some values by substituting them into the polynomial:

- For \( x = 2 \):
\[
(2)^3 - 2(2)^2 - 25(2) + 50 = 8 - 8 - 50 + 50 = 0
\]
So, \( x = 2 \) is a root.

### Step 3: Perform Synthetic Division
We divide \( x^3 - 2x^2 - 25x + 50 \) by \( x - 2 \).

The coefficients of the polynomial are: \( 1, -2, -25, 50 \).

Performing synthetic division:

\[
\begin{array}{r|rrrr}
2 & 1 & -2 & -25 & 50 \\
  &   & 2  & 0   & -50 \\
\hline
  & 1 & 0  & -25 & 0 \\
\end{array}
\]

The quotient is \( x^2 - 25 \) with no remainder, meaning the factorization so far is:
\[
(x - 2)(x^2 - 25) = 0
\]

### Step 4: Factor \( x^2 - 25 \)
The expression \( x^2 - 25 \) is a difference of squares, which factors as:
\[
x^2 - 25 = (x - 5)(x + 5)
\]

### Step 5: Final Factored Form
Thus, the factored form of the polynomial is:
\[
(x - 2)(x - 5)(x + 5) = 0
\]

### Step 6: Complete the Fill-in-the-Blank Questions
- The polynomial equation has at most **3** real roots.
- The leading term is **\( x^3 \)**.
- The leading coefficient is **1**.
- The degree is **3**.
- The roots or zeros of the polynomial are **2, 5**, and **-5**.
- The factors are \( (x - 2), (x - 5), (x + 5) \).

Would you like more details or have any questions?

### Related Questions:
1. How does synthetic division work in polynomial division?
2. What is the Rational Root Theorem, and how is it applied?
3. Can you factor a cubic polynomial using other methods besides synthetic division?
4. What is the difference of squares, and when can it be used in factoring?
5. How do you determine the number of real roots in a polynomial?

### Tip:
When solving cubic equations, always start by checking for simple integer roots using the Rational Root Theorem before proceeding with more advanced methods.

Solve -25x + 50 + x^3 - 2x^2 = 0 and write the polynomial in factored form.

Learn how to solve the cubic polynomial -25x + 50 + x^3 - 2x^2 = 0 using synthetic division and factoring techniques. This detailed solution involves finding roots, using the Rational Root Theorem, and expressing the polynomial in its factored form. Suitable for students in Grades 10-12.

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