Unit 2, Lesson 7
Using Factors and Zeros
Integrated Math 3
Sign in to view assessments and invite other educators
Sign in using your existing Kendall Hunt account. If you don’t have one, create an educator account.
Mai graphs the function given by and sees this graph.
She says, “This graph looks like a parabola, so it must be a quadratic.”
Graph of polynomial function, xy-plane, origin O. Horizontal axis, scale -10 to 8 by 2’s. Vertical axis, scale -40 to 35, by 5’s. Polynomial graph starts in Quadrant 2 and passes down through about (-1 point 7 comma 35), continues passing down through (-1 comma 0) and through (0 comma -30), descending in a smooth curve to a vertex about (point 5 comma -35), then ascending in a smooth curve to the upper right through (2 comma 0) and continues through about (2 point 5 comma 30).
We can use the zeros of a polynomial function to figure out what an expression for the polynomial might be. One way to write a polynomial expression is as a product of linear factors.
For example, for a polynomial function that satisfies when is -1, 2, or 4, we could multiply together a factor that is 0 when , a factor that is 0 when , and a factor that is 0 when . It turns out that there are many possible expressions for .
Using linear factors, one possibility is .
Another possibility is , since the 2 (or any other rational number) does not change what values of make the function equal to 0.
We can test the three values -1, 2, and 4 to make sure that is 0 for those values. We can also graph both possible versions of and see that the graphs intercept the horizontal axis at -1, 2, and 4. Notice that while both functions have the same output at these three specific input values, they have different outputs for all other input values.
Graph of 2 polynomial functions, one solid and one dashed, xy-plane, origin O. Horizontal axis, scale -5 to 4 by 1’s. Vertical axis, scale -20 to 18, by 2’s. Both functions start as increasing in Quadrant 3 and end increasing in Quadrant 1. Both functions have the same roots that are increasing through (-1 comma 0), decreasing through (2 comma 0) and increasing through (4 comma 0). The solid function has a local maximum at (0 comma 8) and a local minimum at (3 comma -4). The dashed function has a local maximum at (0 comma 16) and a local minimum at (3 comma -8).