Question

Write an equation to represent the family of polynomial functions, of degree 4, that have zeros at 3 and 6.

Answer 1

- The question tells us to write the equation that represents a degree 4 polynomial.

- The general form of a degree 4 polynomial is:

`\begin{gathered} f(x)=a(x-\alpha_1)(x-\alpha_2)(x-\alpha_3)(x-\alpha_4) \\ where, \\ \alpha_1,\alpha_2,\alpha_3,\alpha_4\text{ are the roots/zeros of the degree 4 polynomial equation} \\ a\text{ is a constant} \end{gathered}`

- We are told that the zeros are at 3 and 6.

- If this is the case, then, we can proceed to substitute these two numbers into the above formula above.

- The equation can be written in 3 ways:

Scenario 1. When the two zeros, 3 and 4 have the same multiplicity of 2

Scenario 2. When zero of 3 has a multiplicity of 3 and zero of 6 has a multiplicity of 1

Scenario 3. When zero of 6 has a multiplicity of 3 and zero of 3 has a multiplicity of 1

- Thus, we can write the possible scenarios as follows:

`\begin{gathered} Scenario\text{ 1:} \\ f(x)=a(x-3)^2(x-6)^2 \\ \\ Scenario\text{ 2:} \\ f(x)=a(x-3)^3(x-6) \\ \\ Scenario\text{ 3:} \\ f(x)=a(x-3)(x-6)^3 \end{gathered}`