How do you know the multiplicity is 2?

Question

f(x)= x^4 - x^3 - 20x^2

Find all real zeros and determine the multiplicity of each.

i did my factors and got 5,4,1,20,10,2

so i did synthetic with 5 and it was a factor. then I did synthetic with -4 and it was a factor. then x= 0 another zero. And then if I do synthetic again -4 comes out again. My question is how can I know that -4 has a multiplicity of 2 without having to do synthetic to find out so. 

Answer 1

To find the real zeros, set f(x) equal to zero.

 

0 = x⁴ - x³ - 20x²

 

 

Factor out an x² on the right side of equation.  This is the GCF.

 

0 = x²(x² - x - 20)

 

0 = x²(x - 5)(x + 4)

 

 

The zeros are  -4, 0, and 5.   Since the factor x has an exponent of 2, 0 has a multiplicity of 2 because the factor appears twice.   (x - 5) and (x + 4) both have an exponent of 1.  These factors appear only once, so -4 and 5 both have a multiplicity of 1.