find all the zeros for the function y = x^3 - 3x^2 + 25x - 75

To find the zeros of the function \( y = x^3 - 3x^2 + 25x - 75 \), we need to solve the equation:

\[
x^3 - 3x^2 + 25x - 75 = 0
\]

### Step 1: Try Rational Root Theorem
The Rational Root Theorem suggests testing possible rational roots of the form \( \pm \frac{p}{q} \), where \( p \) is a factor of the constant term (-75) and \( q \) is a factor of the leading coefficient (1).

The factors of 75 are: \( \pm 1, \pm 3, \pm 5, \pm 15, \pm 25, \pm 75 \).

We test these values by substituting them into the equation:

#### Test \( x = 3 \):
\[
y = 3^3 - 3(3^2) + 25(3) - 75 = 27 - 27 + 75 - 75 = 0
\]
So, \( x = 3 \) is a root of the equation.

### Step 2: Perform Synthetic Division
Now that we know \( x = 3 \) is a root, we can divide \( x^3 - 3x^2 + 25x - 75 \) by \( x - 3 \) using synthetic division:

\[
\begin{array}{r|rrrr}
3 & 1 & -3 & 25 & -75 \\
  &   & 3  & 0  & 75  \\
\hline
  & 1 & 0  & 25 & 0   \\
\end{array}
\]

This leaves us with the quotient:
\[
x^2 + 25 = 0
\]

### Step 3: Solve the Quadratic Equation
Now solve \( x^2 + 25 = 0 \):
\[
x^2 = -25
\]
\[
x = \pm \sqrt{-25} = \pm 5i
\]

### Step 4: Conclusion
The zeros of the function \( y = x^3 - 3x^2 + 25x - 75 \) are:
\[
x = 3, \quad x = 5i, \quad x = -5i
\]

Let me know if you'd like further details or if you have any questions!

Here are 5 related questions:
1. How can you verify a root is correct for a polynomial?
2. What is the Rational Root Theorem and when is it useful?
3. How do you perform synthetic division with polynomials?
4. What are imaginary numbers, and how are they represented?
5. What is the relationship between the degree of a polynomial and the number of roots?

**Tip:** Always check for real and complex roots when solving polynomials!

Discover how to find the zeros of the polynomial function y = x^3 - 3x^2 + 25x - 75. This step-by-step solution covers the Rational Root Theorem, synthetic division, and solving quadratic equations with complex numbers. Ideal for students in Grades 10-12.

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