What are the zeros of the function, f(x) = x⁴ - 4x³ + 3x², and what are their multiplicities?

We will be using the concept of zeros of a polynomial and their multiplicities to solve this.

Answer: The zeros of the function, f(x) = x⁴ - 4x³ + 3x² are x = 0, 1, and 3 and their multiplicities are 2, 1, and 1 respectively. 

Let's solve this step by step.

Explanation:

Given that, f(x) = x⁴ - 4x³ + 3x²

f(x) = x²(x² - 4x + 3)

Let's check the discriminant of quadratic equation x² - 4x + 3

D = b² − 4ac = (-4)² − 4(1)(3) = 16 - 12 = 4

Hence, both the roots are real.

Roots of quadratic equation ax² + bx + c = 0 are given by: x = [−b ± √(b² − 4ac)] / 2a

Roots of x² - 4x + 3:

x = [-(-4) ± √4] / 2(1) 

x = [4 ± 2] / 2

x = 6/2 , 2/2

x = 3, 1 

Hence, f(x) = x² (x - 3) (x - 1)

Zeros of the function, f(x) = x² (x - 3) (x - 1) are: 

x² = x.x = 0 ⇒ x = 0

(x - 3) = 0 ⇒ x = 3

(x - 1) = 0 ⇒ x = 1

The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity.

The zero associated with this factor, x = 0, has multiplicity 2 because the factor x occurs twice.

The zero associated with this factor, x = 3, has multiplicity 1 because the factor x occurs only once.

The zero associated with this factor, x = 1, has multiplicity 1 because the factor x occurs only once.

Hence, the zeros of the function, f(x) = x⁴ - 4x³ + 3x² are x = 0, 1, and 3 and their multiplicities are 2, 1, and 1 respectively.