To determine the [tex]\(x\)[/tex]-intercepts of the polynomial function [tex]\(f(x) = x^4 - 5x^2\)[/tex], we need to find the values of [tex]\(x\)[/tex] where the function equals zero, i.e., solve [tex]\(f(x) = 0\)[/tex].
Step-by-step solution:
1. Set the function equal to zero:
[tex]\[
x^4 - 5x^2 = 0
\][/tex]
2. Factor the equation:
First, observe that there is a common factor of [tex]\(x^2\)[/tex] in both terms:
[tex]\[
x^2(x^2 - 5) = 0
\][/tex]
3. Set each factor equal to zero:
Solve for [tex]\(x\)[/tex] by setting each factor equal to zero:
- The first factor is [tex]\(x^2 = 0\)[/tex]:
[tex]\[
x = 0
\][/tex]
- The second factor is [tex]\(x^2 - 5 = 0\)[/tex]:
[tex]\[
x^2 = 5
\][/tex]
To solve for [tex]\(x\)[/tex], take the square root of both sides:
[tex]\[
x = \sqrt{5} \quad \text{or} \quad x = -\sqrt{5}
\][/tex]
4. Summarize the solutions:
The solutions from the factors are:
[tex]\[
x = 0, \quad x = \sqrt{5}, \quad x = -\sqrt{5}
\][/tex]
5. Determine the number of unique [tex]\(x\)[/tex]-intercepts:
These solutions are distinct values of [tex]\(x\)[/tex]:
[tex]\[
x = 0, \quad x = \sqrt{5}, \quad x = -\sqrt{5}
\][/tex]
Since each of these values represents a unique [tex]\(x\)[/tex]-intercept, we conclude that there are 3 [tex]\(x\)[/tex]-intercepts.
Thus, the number of [tex]\(x\)[/tex]-intercepts on the graph of the polynomial function is:
[tex]\[ 3 \][/tex]
