Question

Find all zeros to the following polynomial function. f(x)= 5x^4 -10x^3 +4x^2 -8x.

Please answer in detailed steps I’m confused.

Answer 1

Please check the explanation.

Explanation:

Given the function

`f(x)= 5x^4 -10x^3 +4x^2 -8x`

To determine the zeros, set f(x) = 0

`0=\:5x^4\:-10x^3\:+4x^2\:-8x`

switch sides

`5x^4-10x^3+4x^2-8x=0`

as

`5x^4\:-10x^3\:+4x^2\:-8x=x\left(x-2\right)\left(5x^2+4\right)`

so the equation becomes

`x\left(x-2\right)\left(5x^2+4\right)=0`

Using the zero factor principle:

`\mathrm{\quad \:If}\:ab=0\:\mathrm{then}\:a=0\:\mathrm{or}\:b=0\:\left(\mathrm{or\:both}\:a=0\:\mathrm{and}\:b=0\right)`

`x=0\quad \mathrm{or}\quad \:x-2=0\quad \mathrm{or}\quad \:5x^2+4=0`

so solving

• 
  `x=0`

`x-2=0`

• 
  `x=2`

Therefore, the real zeros are:

`x=0` and
`x=2`

`5x^2+4=0` has all the imaginary zeros.

solving

`5x^2+4=0`

`5x^2=-4`

`x^2=-(4)/(5)`

`\mathrm{For\:}x^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}x=√(f\left(a\right)),\:\:-√(f\left(a\right))`

`x=\sqrt{-(4)/(5)},\:x=-\sqrt{-(4)/(5)}`

`x=i(2√(5))/(5),\:x=-i(2√(5))/(5)` ∵
`√(-1)=i`

Therefore, the total zeros are:

`x=0,\:x=2,\:x=i(2√(5))/(5),\:x=-i(2√(5))/(5)`