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.