 Factor the polynomial and use the factored form to find the zeros. (Enter your answers as a comma-separated list. Enter all answers using the appropriate multiplicities.)
P(x) = x^4 − 5x^3 + 4x^2
I found x=0,4,1 however it is saying I do not have the correct number of terms


Let's start by factoring the polynomial correctly and find all the zeros using the factored form.

The given polynomial is:

\[ P(x) = x^4 - 5x^3 + 4x^2 \]

### Step 1: Factor out the greatest common factor (GCF)
First, we notice that \(x^2\) is a common factor in all the terms:

\[
P(x) = x^2(x^2 - 5x + 4)
\]

### Step 2: Factor the quadratic expression
Now, we need to factor \(x^2 - 5x + 4\). We look for two numbers that multiply to 4 and add to -5. These numbers are -4 and -1.

Thus, we can factor the quadratic as:

\[
x^2 - 5x + 4 = (x - 4)(x - 1)
\]

### Step 3: Combine the factored terms
Now, we combine everything:

\[
P(x) = x^2(x - 4)(x - 1)
\]

### Step 4: Find the zeros
To find the zeros, we set \(P(x) = 0\) and solve:

\[
x^2(x - 4)(x - 1) = 0
\]

This gives three factors:
- \(x^2 = 0 \Rightarrow x = 0\) (with multiplicity 2 because of the square),
- \(x - 4 = 0 \Rightarrow x = 4\),
- \(x - 1 = 0 \Rightarrow x = 1\).

### Final Answer:
The zeros of the polynomial are:

\[
x = 0 \ (\text{multiplicity 2}), \ x = 4, \ x = 1
\]

So, the correct answer is: **0, 0, 4, 1**.

---

Would you like further clarification, or do you have any other questions?

### Related Questions:
1. How do you determine the multiplicity of a root from a polynomial's factors?
2. What happens to the graph of a polynomial at a zero with an even multiplicity?
3. Can you explain the process of factoring a quadratic in more detail?
4. How does the degree of a polynomial relate to the number of zeros?
5. What are the possible types of roots a polynomial can have?

### Tip:
Always check for a greatest common factor (GCF) before attempting other factoring techniques.

Factor the polynomial and use the factored form to find the zeros. (Enter your answers as a comma-separated list. Enter all answers using the appropriate multiplicities.) P(x) = x^4 − 5x^3 + 4x^2 I found x=0,4,1 however it is saying I do not have the correct number of terms.

Learn how to factor the polynomial P(x) = x^4 - 5x^3 + 4x^2 and find its zeros with multiplicities. This step-by-step solution covers algebraic concepts, including factoring, the zero product property, and the concept of root multiplicity. Suitable for Grades 9-11.

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