Developing Factoring Muscles
So far i've only covered trinomials with leading coefficient 1, something like x^2 + 13x +42. Now we turn our attention to a trinomial of the sort 2x^2 + 5x + 2. The leading coefficient here is 2. The first step in factoring a trinomial with leading coefficient not equal to 1 is to multiply that coefficient by the last term in the trinomial. For this example the calculation is 2*2=4. We previously were searching for numbers that when multiplied equalled the last term but now that last term 2 is replaced with this new term 4. However we still want the numbers to add up to 5. The numbers here would be 4 and 1 because 4*1=4 and 4+1=5. The answer, though, is not (x+4)(x+1). We have some more work to do.
Let's write the original trinomial in a different way; 2x^2 + 4x + 1x + 2. Notice i simply split 5x into 4x + 1x. Just look at the first pair of terms for a moment. 2x is what's called the greatest common factor of 2x^2 and 4x. That is to say 2 and 4 are both divisible by 2 and both terms have an x. If you extract, or factor, out 2x you will leave an x behind with the first term and a 2 behind with the second term. In other words 2x^2 + 4x = 2x(1x+2). Now look at the second pair of terms. The only common factor there is 1 and the best we can do is say 1x+2=1(1x+2).
At this point we have established 2x^2 + 4x + 1x + 2 = 2x(1x+2) + 1(1x+2). Think of this equation as us taking four terms and making them into two terms. Two terms that share a common factor of 1x+2. 1x+2 can be considered a factor since it is being multiplied in both instances by some other factor. A classic case of taking a new problem and making it look like an old one. We already handled two pairs with each pair having a common factor so let's do it once again. Upon taking out the common factor 1x+2, we have a 2x left in the first term and a 1 left in the second term. This ultimately yields 2x^2 + 4x + 1x +2 = 2x(1x+2) + 1(1x+2) = (1x+2)(2x+1) or 2x^2 + 4x + 1x + 2 = (x+2)(2x+1). Read over this several times and we will try some more soon.