- Set the polynomial \(f(x) = x^4 - x^3 + x^2 - x\) to zero to find the \(x\)-intercepts.
- Factor the polynomial as \(x(x - 1)(x^2 + 1) = 0\).
- Solve for \(x\), identifying two real roots at \(x = 0\) and \(x = 1\), as \(x^2 + 1 = 0\) has no real solutions.
- Conclude that there are two distinct \(x\)-intercepts, which matches option (c). The final answer is \(\boxed{2}\).
Explanation
- Problem Analysis and Objective
To find the \$x\$-intercepts of the function \(f(x) = x^4 - x^3 + x^2 - x\), we need to determine the real values of \(x\) for which the function equals zero. That is, we solve the equation \(f(x) = 0\), or \(x^4 - x^3 + x^2 - x = 0\). This is a quartic polynomial equation, and our goal is to identify all distinct real roots of this equation, as each real root corresponds to an \(x\)-intercept on the graph.
- Factoring the Polynomial
We can simplify the equation by factoring out the common variable \(x\):
\[x(x^3 - x^2 + x - 1) = 0\]
Next, we focus on the cubic polynomial inside the parentheses: \(x^3 - x^2 + x - 1\). We can factor this by grouping the terms in pairs:
\[x^2(x - 1) + 1(x - 1) = 0\]
Since both terms share a common factor of \((x - 1)\), we can factor it out:
\[(x^2 + 1)(x - 1) = 0\]
Combining these, the fully factored form of our original equation is:
\[x(x - 1)(x^2 + 1) = 0\]
- Identifying Real Roots
Now we apply the Zero Product Property, which states that if a product of factors is zero, then at least one of the factors must be zero. This gives us three potential equations to solve:
- \(x = 0\)
- \(x - 1 = 0\), which gives \(x = 1\)
- \(x^2 + 1 = 0\), which gives \(x^2 = -1\)
The first two equations yield the real solutions \(x = 0\) and \(x = 1\). The third equation, \(x^2 = -1\), has no real solutions because the square of any real number is non-negative. Therefore, the only real roots of the polynomial are \(x = 0\) and \(x = 1\).
- Conclusion
Since there are two distinct real roots, \(x = 0\) and \(x = 1\), the graph of the polynomial \(f(x) = x^4 - x^3 + x^2 - x\) crosses or touches the \(x\)-axis at exactly two points: \((0, 0)\) and \((1, 0)\). Thus, there are two \(x\)-intercepts. Comparing this to the given options, we find the correct choice is (c). The final answer is \(\boxed{2}\).
Examples
In engineering and design, finding the \(x\)-intercepts of a function is crucial for determining equilibrium points or moments where a signal crosses a zero threshold. For instance, in an electrical circuit, a function modeling voltage over time might hit zero at specific points, indicating when the current direction changes. Calculating these intercepts allows engineers to analyze the behavior of their systems effectively and ensure they operate within desired parameters.