Question

Solve f(x)=4x5−8x4+8x2−4x

Answer 1

Given:

The function is:

$f(x)=4x^5-8x^4+8x^2-4x$

To find:

The roots of the given equation.

Solution:

We have,

$f(x)=4x^5-8x^4+8x^2-4x$

For roots, $f(x)=0$.

$4x^5-8x^4+8x^2-4x=0$

$4x(x^4-2x^3+2x-1)=0$

$4x((x^4-1)+(-2x^3+2x))=0$

$4x((x^2+1)(x^2-1)-2x(x^2-1))=0$

On further simplification, we get

$4x(x^2+1-2x)(x^2-1)=0$

$4x(x-1)^2(x+1)(x-1)=0$

$4x(x+1)(x-1)^3=0$

Using zero product property, we get

$4x=0$

$x=0$

Similarly,

$x+1=0$

$x=-1$

And,

$(x-1)^3=0$

$x=1$

Therefore, the zeroes of the given function are $-1,0,1$ and the factor form of the given function is $f(x)=4x(x+1)(x-1)^3$.