Question

What are the zeros of the polynomial function: f(x) = (x – 3)(x + 1)(x + 5)?

Answer 1

f(x) = (x – 3)(x + 1)(x + 5)

0 = (x – 3)(x + 1)(x + 5)

(x – 3)(x + 1)(x + 5) = 0

x-3 = 0 or x+1 = 0 or x+5 = 0 ... Use zero product property

x = ??, or x = ??? or x = ???

Answer 2

Zeroes are where our function vanishes, i.e. equals \(0\). Notice we've already factored our function into a product of nice binomial factors; we know that if we multiply things together to get \(0\), at least one of those things must be \(0\) -- right? Think: \(0\times3=0,0\times400=0,\text{etc}\dots\)

Now, we're doing the same thing here; we have \(x-3\), \(x+1\), and \(x+5\) multiplied together to get \(0\) -- which means one of them has to be \(0\). It could be any of them, so we solve for all possible cases:$$x-3=0\\x=3$$but also$$x+1=0\\x=-1$$and even$$x+5=0\\x=-5$$which means we've found three values of \(x\) that make our function equal \(0\). You can verify real quick:$$f(3)=(3-3)(3+1)(3+5)=0\times4\times8=0\\f(-1)=(-1-3)(-1+1)(-1+5)=-4\times0\times4=0\\f(-5)=(-5-3)(-5+1)(-5+5)=-8\times-4\times0=0$$