SOLUTION: find all real zeros(if any) and state the multiplicity of each f(x)=6x^(4)-24x^(3)+24x^(2)

Question 657435: find all real zeros(if any) and state the multiplicity of each f(x)=6x^(4)-24x^(3)+24x^(2)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

To find the zeros we factor the expression. And when we factor we start with the greatest common factor (GCF). The GCF of the right side is :

Next we factor the trinomial. The trinomial fits the patterns of with the "a" being "x" and the "b" being 2:

Now that f(x) is factored we can find the zeros. The zeros of f(x) are the values of x that make f(x) equal to zero. And now that we have f(x) factored (i.e. expressed as a product) we can use the Zero Product Property to find the zeros. Only the values of x that make the factors zero will make f(x) be zero. So our solutions will come from:

and

The first equation in untrue and has no solution. The solution to the second equation is x = 0 and the solution to the third equation is x = 2.

Finally, the multiplicity of each zero is the number of times each zero's factor is a factor of f(x). x = 0 is a zero of f(x) because of the factors of "x". Since f(x) has as a factor, it has two factors of x. So the multiplicity of the zero of 0 is two. Similarly, since there are two factors of (x-2) in f(x) the multiplicity of the zero of 2 is also two.

In summary, the zeros of f(x) are 0 and 2, each with a multiplicity of 2.
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