Find the zeros of each function. State the multiplicity of any multiple zeros.
y = (x-5)(x+2)^2
To find the zeros of the function, we set y equal to zero and solve for x.
0 = (x-5)(x+2)^2
Setting each factor equal to zero:
x - 5 = 0 or (x + 2)^2 = 0
Solving for x in the first equation:
x = 5
Solving for x in the second equation:
(x + 2)^2 = 0
Taking the square root of both sides, we get:
x + 2 = 0
Solving for x:
x = -2
Therefore, the zeros of the function are x = 5 and x = -2.
To determine the multiplicity of each zero, we look at the exponent of each factor.
The zero x = 5 has a multiplicity of 1 because the factor (x - 5) has an exponent of 1.
The zero x = -2 has a multiplicity of 2 because the factor (x + 2) is squared, indicating an exponent of 2.