Question

how many x-intercepts appear on the graph of this polynomial function?
$f(x)=x^4 - 5x^2$
1 x-intercept
2 x-intercepts
3 x-intercepts
4 x-intercepts

Explanation:

Step1: Find x-intercepts (set \( f(x) = 0 \))

Set \( x^4 - 5x^2 = 0 \). Factor out \( x^2 \): \( x^2(x^2 - 5) = 0 \).

Step2: Solve for \( x \)

From \( x^2 = 0 \), we get \( x = 0 \) (a repeated root, but still one unique intercept here? Wait, no—wait, \( x^2 - 5 = 0 \) gives \( x^2 = 5 \), so \( x = \sqrt{5} \) or \( x = -\sqrt{5} \). So the roots are \( x = 0 \), \( x = \sqrt{5} \), \( x = -\sqrt{5} \). Wait, but \( x = 0 \) is a root with multiplicity 2 (from \( x^2 \)), but the x-intercepts are the distinct real roots. Wait, no—wait, the equation \( x^4 - 5x^2 = 0 \) factors to \( x^2(x^2 - 5) = 0 \). So the solutions are \( x = 0 \) (with multiplicity 2) and \( x = \pm\sqrt{5} \) (each with multiplicity 1). But the x-intercepts are the points where the graph crosses or touches the x-axis. The root \( x = 0 \) is a touch point (since multiplicity is even), but it's still one x-intercept? Wait, no—wait, no, the x-intercepts are the distinct real solutions. Wait, \( x = 0 \) is one, \( x = \sqrt{5} \) is another, \( x = -\sqrt{5} \) is the third. Wait, let's solve again:

\( x^4 - 5x^2 = 0 \)

Factor: \( x^2(x^2 - 5) = 0 \)

So either \( x^2 = 0 \) (so \( x = 0 \)) or \( x^2 - 5 = 0 \) (so \( x^2 = 5 \), so \( x = \sqrt{5} \) or \( x = -\sqrt{5} \)).

So the real roots are \( x = 0 \), \( x = \sqrt{5} \), \( x = -\sqrt{5} \). So that's three distinct real roots, hence three x-intercepts. Wait, but the original function is \( f(x) = x^4 - 5x^2 \), which is a quartic. Let's check the graph: at \( x = 0 \), it touches the x-axis (since multiplicity 2, even), and at \( x = \pm\sqrt{5} \), it crosses the x-axis. So the x-intercepts are at \( (0, 0) \), \( (\sqrt{5}, 0) \), and \( (-\sqrt{5}, 0) \). So that's three x-intercepts. Wait, but the options are 1, 2, 3, 4. So the correct answer is 3? Wait, but let's re-express the function: \( f(x) = x^2(x^2 - 5) \). So the roots are \( x = 0 \) (multiplicity 2), \( x = \sqrt{5} \) (multiplicity 1), \( x = -\sqrt{5} \) (multiplicity 1). The x-intercepts are the distinct real roots, so three distinct real roots, so three x-intercepts. Wait, but maybe I made a mistake. Wait, no—wait, the x-intercepts are the number of distinct real solutions. So \( x = 0 \), \( x = \sqrt{5} \), \( x = -\sqrt{5} \): three solutions. So the answer is 3.

Answer:

3 x-intercepts