About Factoring by Grouping

Procedure

• When factoring by grouping, divide a polynomial into two or more groups and remove the greatest common factor from each group. The numbers and variables that are left should be identical, allowing you to simplify the expression.
  
  

Example

• Factor x^3 + 5x^2 + 2x + 10 by grouping it into x^3 + 5x^2 plus 2x + 10. The first half factors to x^2(x + 5) and the second half factors to 2(x + 5). Simplify this answer to (x^2 + 2)(x + 5).
  
  

Identification

• Problems that can be solved using factoring by grouping are typically polynomials with four or more terms because at least two groups of terms must be formed.
  
  

Three-Term Polynomials

• Factoring by grouping is sometimes used to factor three-term polynomials with a first term that has a coefficient greater than one. This process is called "splitting the middle" because the middle term of the polynomial is split to make a four-term polynomial.
  
  

Example

• Factor the three-term polynomial 6x^2 - 7x - 3 by splitting the 7x in a productive way and then factoring by grouping. To find how to split the 7x, multiply the first and last terms, 6x^2 and -3, to get -18x^2. Then find two numbers that add to make 7x and multiply to make -18x^2, which are 9x and -2x. Rewrite the polynomial as 6x^2 - 9x + 7x - 3 and factor by grouping as explained above to get (2x - 3)(3x + 1).