Question

How do you factor completely `2x^4+x^3+2x+1`?

Answer 1

`(2x+1)(x+1)(x-1/2-1/2sqrt3i)(x-1/2+1/2sqrt3i)`

Explanation:

`color(blue)"factor by grouping"`

`=color(red)(x^3)(2x+1)color(red)(+1)(2x+1)`

`"take out the "color(blue)"common factor "(2x+1)`

`=(2x+1)(color(red)(x^3+1))`

`x^3+1" is a "color(blue)"sum of cubes"`

`•color(white)(x)a^3+b^3=(a+b)(a^2-ab+b^2)`

`rArrx^3+1=(x+1)(x^2-x+1)`

`"we can factor "x^2-x+1" by solving "x^2-x+1=0`

`"using the "color(blue)"quadratic formula"`

`"with "a=1,b=-1" and "c=1`

`rArrx=(1+-sqrt(1-4))/2=(1+-sqrt3i)/2=1/2+-1/2sqrt3i`

`(x-(1/2+1/2sqrt3i))(x-(1/2-1/2sqrt3i))`

`=(x-1/2-1/2sqrt3i)(x-1/2+1/2sqrt3i)`

`rArr2x^4+x^3+2x+1`

`=(2x+1)(x+1)(x-1/2-1/2sqrt3i)(x-1/2+1/2sqrt3i)`