How to Solve Polynomials by Grouping

Sometimes the arrangement of a polynomial's terms can be the key to solving it. Grouping is a solving method that searches for patterns within the polynomial by first separating the original equation into smaller equations, or binomials, and then finds the greatest common factor of those binomials. Using grouping to solve polynomials can help break the equation down into smaller and more readily solvable statements.

Instructions

1. Separate the equation into two separate equations by placing parentheses around the first and last pairs of terms. For example, if the equation is x^3-2x^2-5x+10, then placing parentheses around the pairs of expressions yields

   (x^3-2x^2) + (-5x+10).

2. Factor out the greatest common factor out of each of the smaller equations. For the example, the greatest common factor of the first equation is x^2, making the first pair of terms read x^2(x-2), and the greatest common factor of the second equation is -5, resulting in -5(x-2). The full equation now reads x^2(x-2)-5(x-2).

3. Factor out the common equation from the terms. For the example, (x-2) can be factored from the terms, resulting in (x^2-5)(x-2).

4. Set each of the binomials to zero and solve. For x^2-5=0, x^2 equals 5 and x=+√5 and -√5. For x-2=0, x equals 2. The values of x are +√5, -√5 and 2.

References

• West Texas A&M Mathematics Dept: Intermediate Algebra Tutorial 27, The GCF and Factoring by Grouping
• Paul's Online Math Notes: Algebra, Factoring Polynomials