Factoring FOIL expressions is as much an art as it is a science.

First of all, you have to know how to add, subtract, and 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
x^{2}+5x+6

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 1.

**Example 2:** Factor
x^{2}-5x+6

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
x^{2}-x-6

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.

Number #1 | Number #2 | Multiply to | Add to |
---|---|---|---|

-1 | 6 | -6 | 5 |

1 | -6 | -6 | -5 |

-2 | 3 | -6 | 1 |

2 | -3 | -6 | -1 |

As you can see from our table, the numbers are 2 and -3, and our factored expression is (x+2)(x-3).

**Example 4:** Factor
x^{2}-25.

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
x^{2}+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
adding,
subtracting, and
mutiplying integers.
Our factored expression is (x+5)(x-5).

**Example 5:** Factor
-2x^{2}+19x-24.

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 does.

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