How many x-intercepts appear on the graph of this polynomial function?

f (x) = x Superscript 4 Baseline minus x cubed + x squared minus x
1 x-intercept
2 x-intercepts
3 x-intercepts
4 x-intercepts

To find the number of x-intercepts of the polynomial function \( f(x) = x^4 - x^3 + x^2 - x \), we need to determine where the function crosses the x-axis. The x-intercepts occur where \( f(x) = 0 \), so let's solve the equation:

\[
x^4 - x^3 + x^2 - x = 0
\]

### Step 1: Factor the polynomial
First, factor out the common factor \( x \):

\[
x(x^3 - x^2 + x - 1) = 0
\]

Now, we have two factors:

\[
x = 0
\]
\[
x^3 - x^2 + x - 1 = 0
\]

### Step 2: Solve for \( x \)
- One solution is \( x = 0 \), which gives one x-intercept.

- For the cubic factor \( x^3 - x^2 + x - 1 = 0 \), we need to solve this equation. You can try factoring by grouping or applying methods such as the Rational Root Theorem, synthetic division, or numerical approaches, but this equation actually has a real root at \( x = 1 \) (you can verify this by substituting \( x = 1 \) into the cubic).

This means the equation \( x^3 - x^2 + x - 1 = 0 \) has one real solution, \( x = 1 \). Since the cubic doesn't factor into linear real factors, and no other real roots exist, the cubic has only one real root.

### Step 3: Count the x-intercepts
- From \( x = 0 \), we get **one** x-intercept.
- From \( x = 1 \), we get **another** x-intercept.

Thus, the function has **2 x-intercepts**.

### Final Answer:
**2 x-intercepts**

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Would you like more details or have any other questions?

Here are 5 related questions to explore:
1. How do you apply the Rational Root Theorem to solve cubic polynomials?
2. What are the graphical interpretations of x-intercepts in polynomial functions?
3. How can you check for repeated roots in polynomials?
4. How do you solve a cubic equation if factoring fails?
5. How does the degree of a polynomial relate to the number of possible x-intercepts?

**Tip:** Polynomials of degree \( n \) can have up to \( n \) real roots, but some may be complex or repeated.

How many x-intercepts appear on the graph of this polynomial function?

f(x) = x^4 - x^3 + x^2 - x
1 x-intercept
2 x-intercepts
3 x-intercepts
4 x-intercepts

Learn how to determine the number of x-intercepts for the polynomial function f(x) = x^4 - x^3 + x^2 - x through factoring and solving for the roots. This problem involves polynomial functions and factoring methods, suitable for students in Grades 10-12.

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