Problem 64 - College Algebra

Question

Can the graph of a polynomial function have no \(x\) -intercepts? Explain.

Step-by-Step Solution

Answer

Yes, it is possible for the graph of a polynomial function to have no x-intercepts. This can occur if the function represents a constant that is not equal to zero or if the function is shifted vertically in a way that it does not cross the x-axis.

Step 1: Define an x-intercept

An x-intercept is an x-coordinate where the function's value is zero, represented by points on the graph where it crosses the x-axis.

Step 2: Consider a constant polynomial

Consider a constant polynomial function, such as f(x) = c where c is not equal to zero. This function's graph is a horizontal line that never touches or crosses the x-axis, thus it does not have any x-intercepts.

Step 3: Consider Zero Polynomial Function

Consider another scenario where a polynomial function is f(x) = 0, this function has every value of x as its x-intercept.

Step 4: Discuss other polynomial functions

Other polynomial functions of higher degrees (linear, quadratic, cubic, etc.) will generally have one or more x-intercepts, unless they are shifted vertically in such a way that they do not cross the x-axis. Thus, it is possible for a polynomial function to have no x-intercepts, although this is not common.