Question

How do you factor `5x^4 + 2x^2 - 7`?

Answer 1

`5x^4+2x^2-7=(x+1)(x-1)(5x^2+7)`

Explanation:

`5x^4+2x^2-7=5x^4+7x^2-5x^2-7`
because,

`7 xx -5 = -35`

`7 + (-5) = 2`

`=x^2(5x^2+7)-1xx(5x^2+7)`

`=(x^2-1)xx(5x^2+7)`

Further,

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

Thus,

`5x^4+2x^2-7=(x+1)(x-1)(5x^2+7)`

Answer 2

`(x-1)(x+1)(5x^2+7)`

Explanation:

`"let "x^2=u`

`=5u^2+2u-7`

`"factor the quadratic using the a-c method"`

`"the factors of the product "5xx-7=-35`

`"which sum to "+2" are "-5" and "+7`

`"use these factors to split the middle term"`

`5u^2-5u+7u-7larrcolor(blue)"factor by grouping"`

`=color(red)(5u)(u-1)color(red)(+7)(u-1)`

`"take out the "color(blue)"common factor "(u-1)`

`=(u-1)(color(red)(5u+7))`

`"change u back to "x^2`

`=(x^2-1)(5x^2+7)`

`(x^2-1)" is a "color(blue)"difference of squares"`

`=(x-1)(x+1)(5x^2+7)`