Question

in the given graph of a cubic polynomial, what are the number of real zeros and complex zeros, respectively?
a. 1 and 3
b. 2 and 2
c. 2 and 3
d. 1 and 2

Explanation:

Step1: Identify Real Zeros

Real zeros of a polynomial are the x - intercepts (where the graph crosses the x - axis). From the given cubic polynomial graph, we can see that the graph crosses the x - axis at 2 points? Wait, no, wait. Wait, a cubic polynomial has degree 3. Wait, looking at the graph: the graph crosses the x - axis at one point? Wait, no, wait the graph: let's check again. Wait, the graph: when x = 4, it crosses the x - axis, and does it cross at another point? Wait, no, wait the left side: the graph comes from the bottom left, goes up, has a peak, then goes down, then up again. Wait, actually, the graph touches or crosses the x - axis? Wait, no, the graph crosses the x - axis at x = 4 (one point) and does it cross at another? Wait, no, wait the left part: the graph is below the x - axis on the left, comes up, has a maximum, then goes down, then up. Wait, maybe I made a mistake. Wait, a cubic polynomial has degree 3, so the total number of zeros (real and complex) is 3 (by Fundamental Theorem of Algebra). Real zeros are the x - intercepts (where y = 0). From the graph, the graph crosses the x - axis at 1 point? Wait, no, wait the graph: let's look at the x - axis. The graph crosses the x - axis at x = 4 (one point) and does it touch or cross at another? Wait, no, the left side: the graph is below the x - axis, comes up, has a maximum at y negative, then goes down, then up. Wait, maybe the graph crosses the x - axis at 1 point? Wait, no, wait the options: let's recall that for a cubic polynomial, the number of real zeros can be 1 or 3 (since complex zeros come in pairs). Wait, the graph: let's see the x - intercepts. The graph crosses the x - axis at x = 4 (one point) and does it cross at another? Wait, no, maybe I misread. Wait, the graph: when x is negative, the graph is going down, then up, then down, then up. Wait, no, the given graph: the y - axis is vertical, x - axis horizontal. The graph starts from the bottom left (x negative, y negative), goes up, reaches a maximum (at x = 0, y = - 1? Wait, no, the peak is at x = 0, y = - 1? Then goes down, reaches a minimum, then goes up, crossing the x - axis at x = 4. Wait, so the graph crosses the x - axis only at x = 4? No, that can't be. Wait, maybe the graph crosses the x - axis at two points? Wait, no, let's count the number of times the graph intersects the x - axis. Wait, the graph: from the left, comes up, crosses the x - axis? No, when x is very negative, y is negative, then it goes up, has a maximum (y negative), then goes down, has a minimum (y negative), then goes up, crossing the x - axis at x = 4. Wait, so only 1 real zero? Wait, but a cubic must have at least 1 real zero. Then, since the degree is 3, the number of complex zeros (non - real) is 3 - number of real zeros. If real zeros are 1, then complex zeros are 2 (since complex zeros come in conjugate pairs). Let's check the options. Option D: 1 and 2. Let's verify:

1. Real zeros: The number of x - intercepts (where the graph crosses the x - axis) is 1 (only at x = 4, maybe? Wait, no, maybe I made a mistake. Wait, the graph: maybe it crosses the x - axis at two points? Wait, no, looking at the graph again: the left side, the graph is below the x - axis, comes up, has a peak (at x = 0, y = - 1), then goes down, then up, crossing the x - axis at x = 4. So only 1 x - intercept. So real zeros = 1. Then total zeros (real + complex) = 3 (degree 3), so complex zeros = 3 - 1 = 2. So the answer should be D.

Step2: Verify with Fundamental Theorem

The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n zeros (counting multiplicities) in the complex plane. For a cubic polynomial (degree 3), total zeros (real and complex) is 3. Complex zeros (non - real) come in conjugate pairs (so even number). So possible real zeros: 1 or 3 (since 3 is odd, 1 is odd; 2 would be even, but complex zeros come in pairs, so 3 - 2 = 1, which is odd, no, wait: if real zeros are 1, complex zeros are 2 (pair); if real zeros are 3, complex zeros are 0. From the graph, we see 1 x - intercept (real zero), so complex zeros = 3 - 1 = 2. So the number of real zeros is 1, complex zeros is 2.

Answer:

D. 1 and 2