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Applications of the Algebraic Identities

Using the Algebraic Identities in Reverse

It’s very useful to know what (a + b)2 looks like when it has been expanded, but it’s often even more useful to know how to rewrite a2 + 2ab + b2. This means that even though (a + b)2 = a2 + 2ab + b2, it’s just as important to remember the reverse, that a2 + 2ab + b2 = (a + b)2.

Example 1

Factorize x2 + 6x + 9 by applying the first algebraic identity in reverse

You want to end up with an expression of the form (a + b)2, which means you need to find a and b. The expression you have been given is of the form a2 + 2ab + b2. Because the first terms in the two expressions are a2 and x2, a is equivalent to x.

So, you can find b by setting b2 = 9 and checking which values for b yield 2ab = 6x:

9 = ±3, 3 2x = 6x, 3 2x = 6x

9 = ±3,3 2x = 6x, 3 2x = 6x

You can see that b has to be 3, because b = 3 does not match with 2ab = 6x. That gives you x2 + 6x + 9 = x2 + 2 3x + 3 3 = (x + 3)2

Let’s look at an example where you also need to factor out common factors.

Example 2

Factorize 2x3 50x

First, you need to factor out the common factors:

= 2x3 50x = 2 x x x 2 5 5 x = 2x (x2 25) .

2x3 50x = 2 x x x 2 5 5 x = 2x (x2 25) .

The expression (x2 25) is of the form (a2 b2) with a = x and b = 5, which means you can use the third algebraic identity in reverse. That gives you (x2 25) = (x + 5)(x 5), which in turn means that
2x (x2 25) = 2x(x + 5)(x 5)

Simplifying Rational Expressions

If you apply everything you learned about factorization and algebraic identities, you can now simplify fractions that look like this:

x2 + ax + a2 x2 + bx + b2

When you simplify fractions, it’s extremely important to remember that common factors need to be present in all terms. If there’s a term in the fraction without a factor all the other terms have, you can’t cancel that factor! Here are some examples:

Example 3

Simplify the fraction 3x 6 x2 4x + 4

This fraction has two terms in the numerator and three terms in the denominator. The two terms in the numerator are 3x = 3 x and 6 = 2 3. The three terms in the denominator are x2 = x x, 4x = 4 x and 4 = 2 2. There are no factors that are common between all these terms, so you can’t cancel anything as-is.

But, if you factorize the numerator and denominator separately, you’ll see some common factors appearing: Numerator = 3x 6 = 3(x 2) Denominator = x2 4x + 4 = (x 2)2 = (x 2)(x 2)

Now that you’ve factorized the numerator and denominator, you’re left with one term in each: 3(x 2) in the numerator and (x 2)2 in the denominator. You can see that (x 2) is a common factor between the two terms, which means it can be canceled: 3x 6 x2 4x + 4 = 3(x 2) (x 2)2 = 3(x 2) (x 2)(x 2) = 3 x 2

Note! As you can see in this example, a term doesn’t have to be a single variable. It can also be an expression, such as (x 2).

Example 4

Simplify the fraction x + 1 x2 4 + x 2 + 1 4

There’s nothing to be done with the numerator, but if you multiply the denominator by 4, you get 4 (x2 4 + x 2 + 1 4) = x2 + 2x + 1 = (x + 1)2.

This new denominator is much easier to work with. Because you multiplied the denominator by 4, you need to do the same with the numerator. That means the numerator becomes 4 (x + 1). You can see that (x + 1) is a common factor, which gives you

x + 1 x2 4 + x 2 + 1 4 = 4(x + 1) (x + 1)2 = 4 x + 1

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The Third Algebraic Identity
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