Problem 78 - Algebra and Trigonometry

Question

Write equations for several polynomial functions of odd degree and graph each function. Is it possible for the graph to have no real zeros? Explain. Try doing the same thing for polynomial functions of even degree. Now is it possible to have no real zeros?

Step-by-Step Solution

Answer

Odd-degree polynomial functions always have at least one real root as their graph always crosses the x-axis. Even-degree polynomial functions can have no real roots if their graph never intersects the x-axis, such as \(y=x^2+1\). It's the degree of the polynomial that plays a significant role in the graph and roots of the function.

Step 1: Formulating and Graphing Odd-degree Polynomial

Begin by creating equations for odd-degree polynomial functions. Examples are \(y = x^3\) or \(y = x^5\). When graphing these, you'll notice that one end of the function extends to negative infinity and one to positive infinity, regardless of the specifics of the function.

Step 2: Evaluate Odd-degree Polynomial Graphs for Real Roots

The graph of an odd-degree polynomial function always spans from negative infinity to positive infinity or vice versa. This means they always cut the x-axis at least once, hence these functions always have at least one real root.

Step 3: Formulating and Graphing even-degree Polynomial

Next, do the same for even-degree polynomial functions. Examples are \(y = x^2\) or \(y = x^4\). When graphed, you'll notice that both ends point in the same direction (either up or down).

Step 4: Evaluate even-degree Polynomial Graphs for Real Roots

The graph of an even-degree polynomial function either rises on both ends or falls on both ends. It is possible for such a graph to never cross the x-axis, hence it's possible for these functions to have no real roots. An example would be \(y=x^2+1\), which never meets the x-axis thus has no real zeros.