What is the standard form of the polynomial equation given the following zeros: 0, ½, and 3 with a multiplicity of 2?

To find the standard form of the polynomial based on the given zeros 0, ½, and 3 with a multiplicity of 2, we start by constructing the factored form of the polynomial equation.

The zero at 0 implies a factor of x. The zero at ½ implies a factor of (x - ½), and since it has a multiplicity of 2, this factor will be squared. Therefore, we can represent this as (x - ½)². Similarly, the zero at 3 suggests a factor of (x - 3), which also has a multiplicity of 2, leading to (x - 3)².

Combining these, the polynomial in its factored form is:

[

y = x \cdot (x - \frac{1}{2})^2 \cdot (x - 3)^2

]

Next, we can expand each factor. The squared factor (x - ½)² can be expanded as:

[

(x - \frac{1}{2})(x - \frac{1}{2}) = x^2 - x + \frac{1}{4}

]

The square of (x