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 \(K(x)= 3(x+3)(2x-5)\).
- How are the two functions alike? How are they different?
- 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.”
- Is Mai correct? Use graphing technology to check.
- Explain how you could select a viewing window before graphing an expression like \(p(x)\) that would show the main features of a graph.
- 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.
- \((4, 0)\)
- \((0, 0)\) and \((4, 0)\)
- \((\text-2, 0)\), \((0,0)\) and \((4,0)\)
- \((\text-4,0), (0,0)\), and \((2,0)\)
- \((\text-5, 0)\), \(\left(\frac12, 0 \right)\), 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.
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.