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64x³ + 125y³
There are two ways to do this.
The cube root of 64x³ is 4x and the cube root of 125y³ is 5y
You can use long division to divide by the sum (or difference if it
had been the difference of two cubes) of the two cube roots, like this,
1. Divide by the sum of the cube roots of the terms
inserting two placeholder zero middle terms 0x²y + 0xy²
16x² - 20xy + 25y²
4x + 5y)64x³ + 0x²y + 0xy² + 125y³
64x³ + 80x²y
-80x²y + 0xy²
-80x²y - 100xy²
100xy² + 125y³
100xy² + 125y³
0
And so the factorization is
16x³ + 25y³ = (4x + 5y)(64x² - 20xy + 25y²)
Or you can do as most people do -- memorize how the sum or
difference of two cubes factors:
A³ ± B³ = (A ± B)(A² ∓ AB + B²)
The double sign symbols ± and the ∓ mean either use the signs
on top or the signs on the bottom
A will represent the cube root of the first term and B the cube root
of the second term.
Your problem
64x³ + 125y³
The cube root of 64x³ is 4x and the cube root of 125y³ is 5y.
So we substitute 4x for A and 5y for B and ± means + and ∓ means -
So A³ ± B³ = (A ± B)(A² ∓ AB + B²) becomes:
(4x)³ + (5y)³ = (4x + 5y)[(4x)² - (4x)(5y) + (5y)²]
16x³ + 25y³ = (4x + 5y)(64x² - 20xy + 25y²)
Edwin
