Question

Find the zeros of the polynomial function and state the multiplicity of each.

f(x) = 5(x + 7)^2(x - 7)^3

Answer 1

ok, so for what values of x does f(x)=0?

Answer 2

I'll get you started. If x=-7 you have:
f(x)=5(-7+7)^2(x-7)^3
f(x)=5(0)^2(x-7)^3
f(x)=0
Since the exponent on the (x+7) term is 2, we say that this zero has a "multiplicity of 2". You can find the other one I think now.

Answer 3

note: I didn't bother inputting x=-7 to the second term above (it won't change anything anyways).

Answer 4

how did you get (x+7) if the problem is \[f(x) = 5(x + 7)^{2}(x - 7)^{3}\]

don't you have to solve (x+7)(x+7) and then (x-7)(x-7)(x-7) ?

Answer 5

You don't have to expand this one. You just need to see what values of x will make the entire thing zero. If the (x+7)^2 term is zero, then the entire f(x) is zero because zero multiplied by anything is zero. So I noted that if x=-7, then (-7+7)^2=0.
So,
f(-7)=5(0)(-7-7)^3=0
So x=-7 is a zero. It has a multiplicity of two because it appears twice.
f(x)=5(x+7)(x+7)(x-7)^3

Answer 6

Now you have to look at the other term (x-7)^3 to find the other zero.