Question

which statement about the zeros of the graphed function is true? a. the function has three distinct real zeros. b. the function has two distinct real zeros and two complex zeros. c. the function has four distinct real zeros. d. the function has one distinct real zero and two complex zeros.

Explanation:

Step1: Recall Zero Definition

A real zero of a function is where its graph intersects the x - axis.

Step2: Analyze the Graph

Looking at the graph, we can see that the function intersects the x - axis at three distinct points? Wait, no, let's check again. Wait, the graph crosses the x - axis at one point (let's say near x = 1), touches and turns at x = 6 (a repeated root), and crosses at x = 4? Wait, no, let's count the x - intercepts. Wait, the graph intersects the x - axis at three distinct points? Wait, no, the graph: one crossing at around x=1, one crossing at x = 4, and a touch - and - turn at x = 6 (which is a repeated real zero). Wait, but the options: Let's re - examine. The graph of the function: when we look at the x - axis intersections, the function crosses the x - axis at three distinct points? Wait, no, the graph has three x - intercepts? Wait, no, the options: Option A says three distinct real zeros. Wait, but let's think about the multiplicity. Wait, the graph touches the x - axis at x = 6 (so that's a repeated root, but still a real zero), and crosses at x = 4 and another point (around x = 1). Wait, no, maybe I misread. Wait, the graph: let's see the x - axis. The function crosses the x - axis at x≈1, x = 4, and touches at x = 6 (so x = 6 is a real zero with even multiplicity). So in total, how many distinct real zeros? Let's count the distinct x - values where the graph intersects the x - axis. The graph intersects the x - axis at three distinct points? Wait, no, x = 4, x≈1, and x = 6 (but x = 6 is a touch, so it's a real zero). Wait, but the options: Option A: three distinct real zeros. Wait, but maybe the graph has three x - intercepts (one crossing, one crossing, one touching). But a touch at the x - axis is still a real zero (a repeated root). But the question is about distinct real zeros. Wait, the graph: let's look at the options again. Option A: three distinct real zeros. Option C: four distinct real zeros. Wait, maybe I made a mistake. Wait, the graph: let's see the x - axis. The function crosses the x - axis at x = 4, x≈1, and touches at x = 6. Wait, but is there another crossing? Wait, no, the graph as shown: from the left, it comes down, crosses the x - axis near x = 1, goes down, comes up, crosses at x = 4, goes up, touches at x = 6, then goes up. Wait, so that's three x - intercepts (x≈1, x = 4, x = 6), but x = 6 is a touch, so it's a real zero. But are they distinct? x≈1, x = 4, x = 6 are distinct. Wait, but the option A says three distinct real zeros. But wait, maybe the graph has four real zeros? No, the graph only intersects the x - axis at three points (one crossing, one crossing, one touching). Wait, no, maybe I misread the graph. Wait, the original graph: let's see the x - axis. The function: when x is negative, it's above the x - axis, then crosses at x≈1, goes down, comes up, crosses at x = 4, goes up, touches at x = 6, then goes up. So that's three x - intercepts (x≈1, x = 4, x = 6), so three distinct real zeros. But wait, the option A is "The function has three distinct real zeros". But wait, maybe the graph has four real zeros? No, the graph only intersects the x - axis three times (counting the touch as one). Wait, maybe the answer is A. Wait, but let's think about the fundamental theorem of algebra. The degree of the polynomial: if the graph has a local maximum and minimum, let's see the number of turning points. The graph has two turning points (a minimum and a maximum), so the degree is at least 3. But the options: Option A: three distinct real zeros. Option B: two real, two complex. But complex zeros come in pairs, so if there are complex zeros, the number of real zeros would be even or odd depending on the degree. But the graph shows real zeros, so if the degree is 4, then number of real zeros (counting multiplicity) is 4. But distinct real zeros: let's see, the graph touches at x = 6 (so that's a real zero, multiplicity 2), and crosses at x = 4 (multiplicity 1) and x≈1 (multiplicity 1). So distinct real zeros: x≈1, x = 4, x = 6 (three distinct real zeros). So option A is correct. Wait, but maybe I was wrong. Wait, the graph: let's look at the options again. Option A: three distinct real zeros. Option C: four distinct real zeros. Wait, maybe the graph has four x - intercepts. Wait, no, the graph as drawn: from the left, it crosses the x - axis at x≈-1? No, the x - axis is from - 10 to 10. The graph is on the right side (x>0). Wait, maybe the graph crosses the x - axis at x = 1, x = 4, and touches at x = 6, but is there another crossing on the left? No, the graph on the left (x < 0) is above the x - axis. So only three x - intercepts (x>0). So three distinct real zeros. So the correct answer is A.

Answer:

A. The function has three distinct real zeros.