Lesson 7

$M$ 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 $y=K(x)$?

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

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 $f(x)=(x+1)(x-1)(x-2)(x-28)$?

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 $(3,0)$

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.