Algebra GCF and Factoring

Question

How to find the greatest monomial factor of 5a⁷, -30a⁶, 20a⁵, 75a³

 

What are the factors of x³-27?

Answer 1

We find the greatest common factor of the follow terms by examining the constants and exponents of the variable a.  Among the constants, 5, -30, 20, and 75, the greatest common factor of them all is 5.  Among the variables a⁷, a⁶, a⁵, and a³, the greatest common factor is a³.  Therefore overall, the greatest common monominal factor is 5a³.

 

 

To find the factors of (x³ - 27), we either perform long division or synthetic division.  If we get the quotient to be zero, then we have our factors.  The possible factors are (x - 1), (x + 1), (x - 3), (x + 3), (x - 9), and (x + 9).  If we divide (x³ - 27) by any of these listed factors, and get a quotient of zero, then we found our factors.

 

Using synthetic division, the other factor we get is (x² + 3x + 9), a quadratic factor. 

 

x³ - 27 = (x - 3)(x² + 3x + 9)

 

The quadratic factor cannot be factor factored using FOIL.  Therefore, we have our factors.