Question

Find all zeros: x^5-6x^3+5x

Answer 1

On simplifing the given equation we get,

`x^5-6x^3+5x`
`x(x^4-6x^2+5)`

To find the zeros of the equation, consider

`x(x^4-6x^2+5)=0`

we get, x=0 or,

`x^4-6x^2+5=0`

When x=1, we get 1-6+1=5-5=0

Hence x=1 is the zero of the above equation,

To find the factors, we use L division method

we get,

`x^4+0x^3-6x^2+0x+5=0`

The coefficients of x power in decreasing order is 1,0,-6,0,5

we get,

`\begin{gathered} x^4-6x^2+5=0 \\ (x-1)(x^3+x^2-5x-5)=0 \end{gathered}`

x=1 or,

`x^3+x^2-5x-5=0`

When x=-1, we get -1+1+5-5=0

Hence x=-1 is the zero of the equation.

To find the factors, we use L division method

we get,

we get,

`\begin{gathered} x^3+x^2-5x-5=0 \\ (x+1)(x^2-5)=0 \end{gathered}`

we get, x=-1 or,

`\begin{gathered} x^2-5=0 \\ x^2=5 \end{gathered}`
`x=\sqrt[]{5}\text{ or }x=-\sqrt[]{5}`

The zeros of the given equation are,

`x=0,1,-1,\sqrt[]{5},-\sqrt[]{5}`