Factoring Is Just A Form Of Division
My title is the most important statement a person must comprehend in order to be successful at college algebra. Every operation or function in math that does something has something that undoes it. Addition has subtraction and multiplication has division. There are many others but those two are the most elementary. Most everyone reading this has no trouble understanding that to say 4*6 = 24 is equivalent to saying 24/4 = 6. 4 and 6 are called factors of 24 which is to say 24 is divisible by 4 and 6. Multiplication drives the car forward and division drives the car backward. Let's try now to wrap our mind around this concept but with variables and numbers rather than just numbers.
How would we simplify (1+2)*(3+4)? The easiest way to do it is to add in the parentheses first which reduces the problem to 3*7 which is of course 21. But what if we did something awkward like 1*3 + 1*4 + 2*3 + 2*4. This expression simplifies to 3+4+6+8 which also equals 21. This is just one example but you can check with any set of four numbers that it will always come out the same. So now if we stick a variable in each set of those parentheses we know that the first method of adding in the parentheses will not work because we cannot add a number to a variable. That leaves us with the second method. We can use (x+2)*(x+4) as an example. The expression becomes x*x + x*4 + 2*x + 2*4. Now x*4 is the same as 4*x and if we have 4 x's and 2 x's that makes 6 x's. The resulting simplification is x^2 + 6*x +8. What i have just demonstrated is how to multiply two binomials. The binomials in this case are x+2 and x+4. If we look closer we can see easily where the 6 and the 8 in the answer came from. 6 is just 2 and 4 added together and 8 is just 2 and 4 multiplied together.
With this in mind if i present (x+3)*(x+6) as example 2 then we can skip the intermediate step and realize we will end up with x^2 + 9*x + 18. Factoring is WORKING BACKWARDS and working backwards from multiplication is DIVISION. If you are asked to factor x^2 + 9*x + 18 then you are really being asked to factor the number 18 such that the two numbers add up to 9. The ways to factor 18 are 1*18, 2*9, and 3*6. In the blink of an eye we see the numbers on that list that add up to 9 are 3 and 6 which are the numbers in the original example. Want some more? Come back soon.