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Lesson 119: Factoring Polynomials (Part 2 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, be sure to study the previous one. There are some special types of polynomials that are factored in a special way. These patterns need to be memorized.
Our first example is x2 - 25. Notice that nothing can be factored out. There are no factors that are common to both terms. Also notice that this binomial is the difference of two perfect squares. x2 is a perfect square because it has a precise square root, namely x. 25 is a perfect square because it has a square root of 5.
There is a special way of factoring such a binomial. The answer is actually (x - 5)(x + 5). First let's see that it's correct, and then we'll look at a rule. Multiply those two binomials using the FOIL method that you learned Lesson 117. We get x2 + 5x - 5x - 25. The 5x terms cancel out, leaving us with x2 - 25, so it works.
Now let's look at the rule for factoring the difference of two
perfect squares:
a2 - b2
= (a + b)(a - b). This rule needs to be
memorized, but first let's clarify some points.
Remember that a = √(a2) and b is √(a2).
What this really means is that we take the square root of each term,
and then that becomes the a and the b that we use. Let's
try another example: Factor: 4x2 - 49.
In this case, the square root of the first term is 2x. The
square root of the second term 7. That means our factored
answer is
(2x + 7)(2x - 7). Always use FOIL to make sure that your
answer gets you back to the original polynomial.
Now let's look at the square of a sum. This is a bit
trickier. Let's use FOIL on
(x + 4)(x + 4). Note that an alternate way of writing that is
(x + 4)2. It's the square of a sum. Using
FOIL, we get x2 + 4x + 4x + 16. Combining we get x2
+ 8x + 16. What this means is that if we see a polynomial that
fits this pattern, we can factor it into the binomials that we
started with.
A rule is helpful here. a2 + 2ab + b2 = (a + b)(a + b) = (a + b)2. Notice how the polynomial above fits this pattern. Our a is x, and our b is 4. The 2ab term is 2(x)(4), which is 8x. Since it fits the pattern, we can factor it as (a + b)2.
We have one more rule that you need to learn, and it is quite
similar to the rule above:
a2 - 2ab + b2
= (a - b)(a - b) = (a - b)2. This pattern
is the square of a difference. Let's look at a variant of the
polynomial above: x2 - 8x + 16. This fits the
pattern that was just presented. a is x. b is 4.
2ab is 2(x)(4), which is 8x, but it's being subtracted here, just
like in the formula. That means we can factor the polynomial
as (x - 4)(x - 4), or (x - 4)2.
All of this is quite tricky, so don't worry if you are struggling with it. Just do all of the related exercises in your textbook, and keep thinking about each problem until they start to make sense and you can easily see the patterns.
In the next lesson, we'll learn how to factor polynomials that do not fit any of these patterns.
Remember that you can ask a math question if you have additional questions about a topic, or you can contact me if you have any comments or suggestions for this site.