Sum/Difference of Terms

• Determine if the polynomial is a sum or difference of terms. For cubic polynomials, an expression like x^3 -- a^3 factors into (x-a)(x^2 + ax + a^2). This is a difference of cubes. Polynomials of form x^3 + a^3 factor into (x+a)(x^2 -- ax + a^2). This form is a sum of cubes. Similar factoring methods apply to sums and differences of squared terms (quadratics). Quadratics of form x^2 + 2ax + a^2 factor into (x+a)(x+a), while x^2 -- a^2 = (x+a)(x-a).
  
  

Common Factors

• Isolate a common factor. This factoring method is versatile, as it can simplify a polynomial into more familiar forms. The expression 2yx^3 -- 18xy^3 has common factor 2xy. A partial factorization is 2xy(x^2 -- 9y^2). Observe that x^2 -- 9y^2 is a familiar difference of squares. The complete factorization of 2yx^3 -- 18xy^3 is 2xy(x+3y)(x-3y).
  
  

Middle Term Expansion

• Expand middle terms to identify common factors. For instance, 6x^2 -- x -- 35 is not a sum or difference of squares, nor does it have an obvious common factor, as in the previous section. Note that 6x^2 -- x -- 35 = 6x^2 -- 15x + 14x -- 35. Common factors become evident: 6x^2 -- 15x = 3x(2x-5), and 14x-35 = 7(2x-5). Therefore, 6x^2 -- x -- 35=(2x-5)(3x+7).