Question

Explain how to find each root, with its multiplicity, for the polynomial y=x⁵-27x²?

Answer 1

`y=x^5-27x^2`

In order to find each root we need to know when:

`y=0`

so:

`\begin{gathered} x^5-27x^2=0 \\ \end{gathered}`

Factor x² out of x⁵- 25x²:

`\begin{gathered} x^2(x^3-27) \\ where \\ x^3-27=x^3-3^3 \end{gathered}`

Factor the difference of two cubes:

`\begin{gathered} x^3-3^3=(x-3)(x^2+3x+3^2) \\ so\colon \\ x^2(x-3)(x^2+3x+9) \\ \end{gathered}`

Therefore, the real roots are:

`\begin{gathered} x=0 \\ or \\ x=3 \end{gathered}`

Where the root:

`x=0`

has a multiplicity of 2