Factoring FOIL expressions is as much an art as it is a science.
First of all, you have to know how to
multiply integers. Otherwise,
you are dead in the water. The easiest way to show how to do this is to
work out a few examples.
Example 1: Factor
This is pretty basic. We are looking for two numbers that add to 5
and multiply to 6. Of course, the numbers are 2 and 3. So our factored
expression is (x+2)(x+3). The numbers you would
put in the four solution boxes are 1,2,1,3 because the
coefficient in front of the x is understood to be
Example 2: Factor
This time, we are looking for two numbers that add to -5
and multiply to 6. The numbers are -2 and -3, and our factored
expression is (x-2)(x-3).
Example 3: Factor
This time, we are looking for two numbers that add to -1 (because
the coefficient in front of the x is understood
to be -1) and multiply to -6. If it isn't obvious what the solution is,
the easiest strategy is to write out the number combinations that multiply
to -6 and see if they add up to -1.
As you can see from our table, the numbers
are 2 and -3, and our factored expression is
Example 4: Factor
What we have here is a classic
difference of squares
problem. How can we tell? Because the x term is missing.
Actually, it's not really missing. It's just that the
coefficient is 0.
We could write the expression as
x2+0x-25. We are looking for 2
numbers that multiply to -25 and add to 0. If it isn't obvious that the
numbers are 5 and -5, you really need to practise
Our factored expression is (x+5)(x-5).
Example 5: Factor
There are all sorts of ways to tackle a problem like this. Most involve
factoring and adding/multiplying combinations. The easiest one for my
money is using the quadratic formula. In this case,
we have a = -2,
b = 19, and
c = -24. One root is 8, the other is -3/2.
So, one term is (x-8) and the other is
(-2x+3). Our factored expression is
(x-8)(-2x+3). We just have to make sure
that our factored expression expands to the original problem, which it