What is the factored form of 8x^24-27y^6

The answer will be (2x8 - 3y2) • (4x16 + 6x8y2 + 9y4)

8x24-27y6 Final result : (2x8 - 3y2) • (4x16 + 6x8y2 + 9y4)

Step by step solution : Step 1 : Skip Ad

Equation at the end of step 1 : (8 • (x24)) - 33y6 Step 2 : Equation at the end of step 2 : 23x24 - 33y6 Step 3 : Trying to factor as a Difference of Squares:

3.1 Factoring: 8x24-27y6

Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)

Proof : (A+B) • (A-B) =

A2 - AB + BA - B2 =

A2 - AB + AB - B2 =

A2 - B2

Note : AB = BA is the commutative property of multiplication.

Note : - AB + AB equals zero and is therefore eliminated from the expression.

Check : 8 is not a square!

Ruling : Binomial can not be factored as the

difference of two perfect squares

Trying to factor as a Difference of Cubes:

3.2 Factoring: 8x24-27y6

Theory : A difference of two perfect cubes, a3 - b3 can be factored into

(a-b) • (a2 + ab + b2)

Proof : (a-b) • (a2+ab+b2) =

a3 + a2b + ab2 - ba2 - b2a - b3 =

a3 + (a2b - ba2) + (ab2 - b2a) - b3 =

a3 + 0+0 + b3 =

a3 + b3

Check : 8 is the cube of 2

Check : 27 is the cube of 3

Check : x24 is the cube of x8

Check : y6 is the cube of y2

Factorization is:

(2x8 - 3y2) • (4x16 + 6x8y2 + 9y4)

Trying to factor as a Difference of Squares:

3.3 Factoring: 2x8 - 3y2

Check : 2 is not a square!

Ruling : Binomial can not be factored as the

difference of two perfect squares

Trying to factor a multi variable polynomial:

3.4 Factoring 4x16 + 6x8y2 + 9y4

Try to factor this multi-variable trinomial using trial and error

Factorization fails

Final result : (2x8 - 3y2) • (4x16 + 6x8y2 + 9y4)