# Lesson 7

Using Factors and Zeros

• Let’s write some polynomials.

### 7.1: More Than Factors

$$M$$ and $$K$$ are both polynomial functions of $$x$$ where $$M(x)=(x+3)(2x-5)$$ and $$K(x)= 3(x+3)(2x-5)$$.

1. How are the two functions alike? How are they different?
2. If a graphing window of $$\text-5 \leq x \leq 5$$ and $$\text-20 \leq y \leq 20$$ shows all intercepts of a graph of $$y=M(x)$$, what graphing window would show all intercepts of $$y=K(x)$$?

### 7.2: Choosing Windows

Mai graphs the function $$p$$ given by $$p(x)=(x+1)(x-2)(x+15)$$ and sees this graph.

She says, “This graph looks like a parabola, so it must be a quadratic.”

1. Is Mai correct? Use graphing technology to check.
2. Explain how you could select a viewing window before graphing an expression like $$p(x)$$ that would show the main features of a graph.
3. Using your explanation, what viewing window would you choose for graphing $$f(x)=(x+1)(x-1)(x-2)(x-28)$$?

Select some different windows for graphing the function $$q(x) = 23(x-53)(x-18)(x+111)$$. What is challenging about graphing this function?

### 7.3: What’s the Equation?

Write a possible equation for a polynomial whose graph has the following horizontal intercepts. Check your equation using graphing technology.

1. $$(4, 0)$$
2. $$(0, 0)$$ and $$(4, 0)$$
3. $$(\text-2, 0)$$, $$(0,0)$$ and $$(4,0)$$
4. $$(\text-4,0), (0,0)$$, and $$(2,0)$$
5. $$(\text-5, 0)$$, $$\left(\frac12, 0 \right)$$, and $$(3,0)$$

### Summary

We can use the zeros of a polynomial function to figure out what an expression for the polynomial might be.

Let’s say we want a polynomial function $$Z$$ that satisfies $$Z(x)=0$$ when $$x$$ is -1, 2, or 4. We know that one way to write a polynomial expression is as a product of linear factors. We could write a possible expression for $$Z(x)$$ by multiplying together a factor that is zero when $$x=\text-1$$, a factor that is zero when $$x=2$$, and a factor that is zero when $$x=4$$. Can you think of what these three factors could be?

It turns out that there are many possible expressions for $$Z(x)$$. Using linear factors, one possibility is $$Z(x)=(x+1)(x-2)(x-4)$$. Another possibility is $$Z(x)=2(x+1)(x-2)(x-4)$$, since the 2 (or any other rational number) does not change what values of $$x$$ make the function equal to zero.

To check that these expressions match what we know about $$Z$$, we can test the three values -1, 2, and 4 to make sure that $$Z(x)$$ is 0 for those values. Alternatively, we can graph both possible versions of $$Z$$ and see that the graphs intercept the horizontal axis at -1, 2, and 4, as shown here.