Question

How do you factor `54x^3 - 2y^3`?

Answer 1

`2(3x-y)(9x^2+3xy+y^2)`

Explanation:

When factoring,
Step 1: Always factor out a GCF first, if possible.

In this particular problem, we can factor out 2:
`54x^3-2y^3` = `2(27x^3-y^3)`

Step 2: Count the number of terms:

—if 2 terms, check if problem is:

difference of squares `(a^2-b^2)`, which factors into `(a-b)(a+b)`

difference of cubes `(a^3-b^3)`, which factors into `(a-b)(a^2+ab+b^2)`

sum of cubes `(a^3+b^3)`, which factors into `(a+b)(a^2-ab+b^2)`

(`27x^3-y^3)` is a difference of cubes:
`27x^3` = `(3x)^3`
`y^3` = `(y)^3`

Plugging these values into the difference of cubes formula:
`(a^3-b^3)` = `(a-b)(a^2+ab+b^2)`
`((3x)^3-((y)^3)` = `((3x)-(y)) ((3x)^2+(3x)(y)+ (y)^2)`
`27x^3-y^3` = `(3x-y)(9x^2+3xy+y^2)`

Putting it all together:

`54x^3-2y^3` = `2(27x^3-y^3)`
`54x^3-2y^3` = `2(3x-y)(9x^2+3xy+y^2)`