Question

How do you factor `20x^4+16x^3-5x-4`?

Answer 1

`20x^4+16x^3-5x-4 = (4x^3-1)(5x+4)`

`color(white)(20x^4+16x^3-5x-4) = (root(3)(4)x-1)(2root(3)(2)x^2+root(3)(4)x+1)(5x+4)`

Explanation:

Notice that the ratio of the first and second terms is the same as that of the third and fourth terms. So this quadrinomial will factor by grouping:

`20x^4+16x^3-5x-4 = (20x^4+16x^3)-(5x+4)`

`color(white)(20x^4+16x^3-5x-4) = 4x^3(5x+4)-1(5x+4)`

`color(white)(20x^4+16x^3-5x-4) = (4x^3-1)(5x+4)`

We can factor `4x^3-1` as a difference of cubes using irrational coefficients:

`a^3-b^3 = (a-b)(a^2+ab+b^2)`

So:

`4x^3-1 = (root(3)(4)x)^3-1^3`

`color(white)(4x^3-1) = (root(3)(4)x-1)((root(3)(4)x)^2+root(3)(4)x+1)`

`color(white)(4x^3-1) = (root(3)(4)x-1)(root(3)(16)x^2+root(3)(4)x+1)`

`color(white)(4x^3-1) = (root(3)(4)x-1)(2root(3)(2)x^2+root(3)(4)x+1)`