How to Split the Middle Term?

Polynomials that have their middle term coefficient other than one are supposed to be factored by splitting the term into smaller ones. One way to do is by just making a prediction by judging two Numbers that would give the value of constant 'C' as their product and Coefficient 'b' as their sum. This technique is mostly used in case of quadratic equations with coefficient of higher degree term i.e. 'A' equals to 1. So, how to split the middle term when coefficient A is not equals to 1?

First arrange the terms of polynomial such that it represents the general form of it ax² + bx + c. Let us consider an example of following polynomial equation 2x² + x = + 6.

Polynomial in its standard form would look like 2x2 + x – 6 = 0.

Next we multiply the coefficients of terms x² and x⁰  i.e. “a” and “c” respectively. In our example we would get a * c = 2 * - 6 = - 12. Next we guess two numbers that would result into - 12 as their product and would sum up to value of coefficient of 'x' i.e. b = 1. Two numbers can be 4 and – 3. These numbers when multiplied give us -12 and when added give us 1. So middle term can be split as follows:

2x² + x – 6 = 0,
Or      2x² + 4 x – 3 x – 6 = 0,

Now we can factor our equation by taking common terms as follows:
Or 2 x (x + 2) – 3 (x + 2) = 0,
Or (2 x – 3) (x + 2) = 0,
Or x = 3 /2, -2.