Algebra II For Dummies. Sterling Mary Jane
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Any discussion of algebra presumes that you’re using the correct notation and terminology. Algebra I (check out Algebra I For Dummies [Wiley]) begins with combining terms correctly, performing operations on signed numbers, and dealing with exponents in an orderly fashion. You also solve the basic types of linear and quadratic equations. Algebra II gets into more types of functions, such as exponential and logarithmic functions, and topics that serve as launching spots for other math courses.
Going into a bit more detail, the basics of algebra include rules for dealing with equations, rules for using and combining terms with exponents, patterns to use when factoring expressions, and a general order for combining all the above. In this chapter, I present these basics so you can further your study of algebra and feel confident in your algebraic ability. Refer to these rules whenever needed as you investigate the many advanced topics in algebra.
Outlining Algebraic Properties
Mathematicians developed the rules and properties you use in algebra so that every student, researcher, curious scholar, and bored geek working on the same problem would get the same answer – no matter the time or place. You don’t want the rules changing on you every day (and I don’t want to have to write a new book every year!); you want consistency and security, which you get from the strong algebra rules and properties that I present in this section.
If you add 2 and 3, you get 5. If you add 3 and 2, you still get 5. If you multiply 2 times 3, you get 6. If you multiply 3 times 2, you still get 6.
Algebraic expressions usually appear in a particular order, which comes in handy when you have to deal with variables and coefficients (multipliers of variables). The number part comes first, followed by the letters, in alphabetical order. But the beauty of the commutative property is that 2xyz is the same as x2zy. You have no good reason to write the expression in that second, jumbled order, but it’s helpful to know that you can change the order around when you need to.
You can use the associative property of addition or multiplication to your advantage when simplifying expressions. And if you throw in the commutative property when necessary, you have a powerful combination. For instance, when simplifying (x + 14) + (3x + 6), you can start by dropping the parentheses (thanks to the associative property). You then switch the middle two terms around, using the commutative property of addition. You finish by reassociating the terms with parentheses and combining the like terms:
The steps in the previous process involve a lot more detail than you really need. You probably did the problem, as I first stated it, in your head. I provide the steps to illustrate how the commutative and associative properties work together; now you can apply them to more complex situations.
For instance, you can use the distributive property on the problem
to make your life easier. You distribute the 12 over the fractions by multiplying each fraction by 12 and then combining the results:
Finding the answer with the distributive property is much easier than changing all the fractions to equivalent fractions with common denominators of 12, combining them, and then multiplying by 12.
Applying the additive identity
One situation that calls for the use of the additive identity is when you want to change the format of an expression so you can factor it. For instance, take the expression x2 + 6x and add 0 to it. You get x2 + 6x + 0, which doesn’t do much for you (or me, for that matter). But how about replacing that 0 with both 9 and –9? You now have x2 + 6x + 9 – 9, which you can write as (x2 + 6x + 9) – 9 and factor into (x + 3)2 – 9. Why in the world do you want to do this? Go to Chapter 11 and read up on conic sections to see why. By both adding and subtracting 9, you add 0 – the additive identity.
Making multiple identity decisions
You use the multiplicative identity extensively when you work with fractions. Whenever you rewrite fractions with a common denominator, you actually multiply by one. If you want the fraction
Now you’re ready to rock and roll with a fraction to your liking.
You face two types of inverses in algebra: additive inverses and multiplicative inverses. The additive inverse matches up with the additive identity and the multiplicative inverse matches up with the multiplicative identity. The additive inverse is connected to zero, and the multiplicative inverse is connected to one.