Question

Find the zeroes state the multiplicity of multiple zeroes Y=x^2(x-1)^2(x+3)^5

Answer 1

Given the following function:

$Y\text{ = f(x) = }x^2(x-1)^2(x+3)^5$

Let's first spread the equation based on the exponents,

$\text{ }x^2(x-1)^2(x+3)^5\text{ = x }\cdot\text{ x }\cdot\text{ (x - 1) }\cdot\text{ (x - 1) }\cdot(x+3)\cdot(x+3)\cdot(x+3)\cdot(x+3)\cdot(x+3)_{}$

Let's then find the zeros of the factors,

x → x = 0

x → x = 0

x - 1 → x = 1

x - 1 → x = 1

x + 3 → x = -3

x + 3 → x = -3

x + 3 → x = -3

x + 3 → x = -3

x + 3 → x = -3

Therefore, the zeros are: 0, 0, +1, +1, -3, -3, -3, -3, -3

Let's now determine the multiplicity of multiple zeros,

0 mult. 2

+1 mult. 2

-3 mult. 5