How to Factor Polynomials in Pairs

Factoring in pairs can solve certain polynomials of degree 3 or higher.

Factoring in pairs, also known as factoring by grouping, is an effective method for solving more complex polynomials, but it requires more computation than basic factoring techniques. The goal of factoring in pairs is to split the polynomial into two separate polynomials that share a common factor. The pair method can solve many polynomials other techniques cannot solve.

Instructions

1. Split the polynomial into two separate polynomials of equal length -- this method does not work with polynomials that have an odd number of terms. For example, you could split the polynomial "x^3 + 4x^2 - 3x -12" into "x^3 + 4x^2" and "-3x - 12."

2. Find the greatest common factor (GCF) of both polynomials. The greatest common factor of "x^3 + 4x^2" is x^2 and the greatest common factor of "-3x - 12" is -3.

3. Find the paired factor of each polynomial by dividing it by its GCF. For example, the paired factor of the first polynomial is "x + 4" because x^2 times (x + 4) is "x^3 - 4x^2," the unfactored polynomial. The paired factor of the second polynomial is also "x + 4" because -3 times (x + 4) is "-3x - 12."

4. Check to make sure that the paired factor is the same for both polynomials. If this is not the case, go back to Step 1 and split the polynomial differently, pairing the first and third polynomial together rather than the first and second. If neither pairing works, factoring in pairs may not be the correct approach.

5. Rewrite the polynomial in factored form by multiplying the GCF and paired factor, then recombining the two polynomials. The above example would be rewritten as x^2*(x + 4) - 3*(x + 4).

6. Add the two GCFs together to make one factor of the polynomial. In the example, "x^2" plus "-3" is "x^2 - 3." The other factor is the paired factor of both polynomials, in this case (x + 4). The factored form of the polynomial is the product of these two factors -- (x^2 - 3)*(x + 4).

Tips & Warnings

• If one of your paired factors is the negative version of your other factor (for example, "x + 4" and (-x - 4), you can make the factor positive by changing the sign of the GCF. This will change the sign of every term in its paired factor, turning the negative version into the positive version.

References

• UNT Department of Mathematics: Factoring by Grouping
• Connexions: Factoring Polynomials -- Factoring by Grouping