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Lesson 120: Factoring Polynomials (Part 3 of 4)
In this lesson we'll learn more about the very important topic of factoring polynomials. Be sure to fully understand it before moving on.
Before starting this lesson, study the previous two. In this lesson we'll learn what to do when we have a polynomial that does not fit any of the patterns that we previously saw. Let's take this example: x2 + 7x + 12. Upon inspection, we can see that this polynomial doesn't fit any of the patterns that we've seen. We cannot factor it as the square of a sum or difference. What we need to do is sometimes referred to as "reverse FOIL." We're going to figure out what binomials, when FOIL'd, will give us the original polynomial.
Here is how we start. Our first term is x2. That means that our binomials must be of the form (x )(x ), so that when we multiply the first terms, we'll get x2. Now we should look at the +12. When we multiply the last two terms of our binomials, we need for them to multiply to +12. What we should do is make a list of all the pairs of factors which multiply to +12. They are 1 times 12, 2 times 6, and 3 times 4. Remember that the negatives of each of these will also work. For example -3 times -4 equals +12. So we have 6 possible pairs. Later you'll be able to do all this in your head, but for now, it's important to list them.
Now we need to get our middle term of 7x. To get this, we'll be adding the inner product, and the outer product of our binomials. The inner product will be one of our factors of 12, multiplied by x, and the outer product will be the other factor of 12, multiplied by x. Then we'll be adding those to products together, inner and outer.
What this all means is that we need two numbers, such that their product is +12, and their sum is +7. If we look at our possible factor pairs, we can see that only +3 and +4 works. Our factored answer then becomes (x + 3)(x + 4). Always use FOIL to check that your answer does multiply out to be the same as the original polynomial.
Let's try another one: x2 - 2x - 15. Again, we need two numbers that multiply to +15, but add to -2. Be very careful with your signs. That is the biggest mistake student make with these problems. Let's list pairs of numbers that multiply to -15: (-1, 15), (-3, 5), (1, -15), and (3, -5). Which of these pairs will add to -2? The answer is 3 plus -5. That means our factored answer is (x + 3)(x - 5). Again, use FOIL to make sure that when we multiply the binomials, we get back to our original polynomial.
Doing this takes lots of practice, and it will get easier over time. Eventually the answer will just pop out at you, without much need for computation. Practice these types of problems as much as you can, and always be very careful with your signs and with your basic arithmetic. Don't do 3 - 5 and end up with positive 2, or the whole problem will be wrong.
Now let's try a bit of a different example: 3x2 + 8x + 5. Here we have a constant before the x, so that makes it a bit trickier. First of all, our factored answer will be of the form (3x )(x ). That is the only way that we can get 3x2 when we multiply our first terms. Now the tricky part. We again need a pair of numbers that multiply to +5. However, we need to pick them such that when we multiply our inner and outer terms, and then add those sums together, we get 8x. The problem is that our outer term involves multiplying by 3x, so we must take that into account.
As before, let's list the pairs of numbers that multiply to +5. We only have 1 times 5, and -1 times -5. Now we have to experiment to see not only which pair we should pick, but which of the numbers should go with the 3x, and which should go with the x. Eventually you'll be able to do these problems in your head, but for now, you really just need to experiment and see what works.
First I'll try (3x - 1)(x - 5). I'll use FOIL to see if that gets us the original polynomial. The problem is that the sum of the inner and outer products is -16x, and we need +8x. Let's try the opposite order: (3x - 5)(x - 1). Now our sum of inner and outer is -8x, which is still not right. We'll need to experiment using the pair of 1 and 5, in each of the two possible positions. It turns out that the answer which works is (3x + 5)(x + 1). If we use FOIL, we'll get our original polynomial.
These problems just take a lot of practice and careful work. It's also important to be patient with yourself while you work through them. You'll have more practice with this later, and you'll learn more about factoring in the next lesson.
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