How do you find the zeroes of a function and their multiplicities?

Y= (x-2)^3(2x+5)^2

If (ax-b)=0, then x = b/a

If (ax-b)^n is a factor, then the root has multiplicity n.

So, your roots are
x = 2, multiplicity=3
x = -5/2, multiplicity=2

To find the zeroes of a function and their multiplicities, we need to set the function equal to zero and solve for the values of x. In this case, the given function is:

Y = (x-2)^3 * (2x+5)^2

Step 1: Set the function equal to zero:

0 = (x-2)^3 * (2x+5)^2

Step 2: Apply the zero product property, which states that if a product of factors is equal to zero, then at least one of the factors must be equal to zero:

(x-2)^3 = 0 or (2x+5)^2 = 0

Step 3: Solve for x in each equation:

For (x-2)^3 = 0:
(x-2) = 0
x = 2

For (2x+5)^2 = 0:
Take the square root of both sides to eliminate the square:
2x + 5 = 0
2x = -5
x = -5/2

Step 4: Determine the multiplicities of the zeroes.

The multiplicity of a zero refers to the number of times a particular factor is repeated. In this case, the factors are (x-2) and (2x+5). The exponents indicate the multiplicities.

For (x-2)^3, the multiplicity is 3.
For (2x+5)^2, the multiplicity is 2.

So, the zeroes of the function and their multiplicities are:
x = 2 with multiplicity 3
x = -5/2 with multiplicity 2