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Factoring a Sum or Difference of Two Cubes

Factor: 27w³ + 125y¹²

Solution

Step 1 Decide if the given polynomial fits a pattern.

The first term, 27w³, is a perfect cube, (3w)³.

The last term, 125y¹², is a perfect cube, (5y⁴)³.

The terms are added.

Therefore, 27w³ + 125y¹² is a sum of two cubes.

Step 2 Identify a and b. Then substitute in the pattern and simplify. In the factoring pattern for a sum of two cubes, substitute 3w for a and 5y⁴ for b.

Simplify.

a³ + b³

(3w)³ - (5y⁴)³

= (a + b)(a² - ab + b²)

= (3w - 5y⁴)[(3w)² - 3w Â· 5y⁴ + (5y⁴)²]

= (3w - 5y⁴)(9w² - 15y⁴ + 25y⁸)

The result is: 27w³ + 125y¹² = (3w - 5y⁴)(9w² - 15y⁴ + 25y⁸).

You can multiply to check the factorization. We leave the check to you.