Factorize 125+8x^3+90xy-27y^3
I tried till :
5^3+(2x)^3-9y(10x+3y^2)
got stuck here as there is no commom cuming 5 and 2x n last sction.
I see a lot of perfect cubes there
8x^3 - 27y^3 = (2x-3y)(4x^2+6xy+9y^2)
Hmmm. No joy there.
wolframalpha.com factors it, but I can see no motivation for trying (2x-3y+5) as a factor.
To factorize the expression 125 + 8x^3 + 90xy - 27y^3, you need to look for common factors among the terms. Let's break it down step by step:
1. Notice that each term has a power of 3:
125 = 5^3
8x^3 = (2x)^3
-27y^3 = -(3y)^3
2. Now, observe that the coefficient in front of each term is a perfect cube as well:
125 = 5^3
8 = 2^3
-27 = -(3^3)
3. Next, look for any common variables. In this case, both 8x^3 and 90xy contain the factor x, and -27y^3 and 90xy contain the factor y.
4. Reorganize the terms to group the common factors and rewrite the expression using the pattern (a^3 + b^3):
(5^3 + 2x^3) + (90xy - 27y^3)
5. Factorize each grouped term separately:
5^3 + 2x^3 = (5 + 2x)(25 - 10x + 4x^2)
90xy - 27y^3 = 9y(10x - 3y^2)
6. Combine the factorized terms:
(5 + 2x)(25 - 10x + 4x^2) + 9y(10x - 3y^2)
This is the factorized form of the expression 125 + 8x^3 + 90xy - 27y^3.