We can actually use the Zeros Theorem and the Conjugate Zeros Theorem together to conclude that an odd-degree polynomial with real coefficients must have atleast one real root (since the non-real roots must come in conjugate pairs). But how can we get the same conclusion by just considering the end behavior of an odd-degree polynomial?

Hint: Think about how the root(s) of a polynomial show up on its graph. Then, look at what we know about the end behavior of an odd-degree polynomial, and note what has to happen between the ends.

Doesn't the graph of an odd-degree polynomial with a positive coefficient end up getting lost way up in the first quadrant?

And get lost way down in the third quadrants?

How did it get from way down there in the third to way up there in the first ??
My guess would be it "must have crossed the x-axis somewhere" at least once", meaning, there has to be at least one real root.

If the coefficient is negative, just reverse the logic , "goes from second quadrant to fourth".

Odd degree polynomials graph with the - and + ends on the opposite side of the x axis, don't they. So they have to cross that real axis at least once.

To understand how we can conclude that an odd-degree polynomial with real coefficients must have at least one real root by considering its end behavior, let's first review some key concepts.

The end behavior of a polynomial refers to how the polynomial behaves as x approaches positive or negative infinity. For an odd-degree polynomial, the end behavior can be classified into two scenarios:

1. When the leading coefficient is positive: In this case, as x approaches negative infinity, the polynomial also approaches negative infinity, and as x approaches positive infinity, the polynomial approaches positive infinity.

2. When the leading coefficient is negative: In this case, as x approaches negative infinity, the polynomial approaches positive infinity, and as x approaches positive infinity, the polynomial approaches negative infinity.

Now, let's consider what happens between the ends of an odd-degree polynomial. Since the end behavior of an odd-degree polynomial goes towards opposite infinities on either side, the polynomial must cross the x-axis at least once. This implies that the polynomial must have at least one real root.

To visualize this, imagine graphing an odd-degree polynomial. The graph will show the polynomial approaching opposite infinities on either side. As it does so, it must intersect the x-axis, indicating the presence of at least one real root.

Therefore, by analyzing the end behavior of an odd-degree polynomial, we can conclude that it must have at least one real root. This observation aligns with the conclusions obtained using the Zeroes Theorem and the Conjugate Zeros Theorem, where we determine that non-real roots must come in conjugate pairs.