In the English language, we've got phrases and we've got sentences. A phrase is a string of words that expresses a thought but doesn't form a complete sentence. A sentence is a grammatically correct string of words that includes both a subject and a verb. This is a sentence. Phrase over here.
1. "the fat cat"
2. "purple feathers"
3. "upon the mountain"
4. "the bald eagle"
1. "The fat cat liked mice."
2. "The bird has purple feathers."
3. "The hermit lived upon the mountain."
4. "The bald eagle was disappointed to find that the Rogaine wasn't working."
Not every string of words is a phrase or sentence. Some strings of words are nonsense. Like a good majority of the poppycock that comes twaddling out of our claptraps.
1. "fish dog cat and the"
2. "like swim eat chocolate"
3. "so if but or to"
We apologize if any of these look suspiciously like the text messages you send on a daily basis. We also apologize if the non-phrase "like swim eat chocolate" is ever the heading on your future online dating profile.
In algebra, instead of phrases and sentences, we have expressions and equations. An expression is a string of mathematical symbols representing a quantity. An equation is a string of mathematical symbols stating the equality of two expressions. Therefore, an expression is like a phrase and an equation is like a sentence. Instead of subjects and verbs, we have constants and variables. Instead of punctuation, we have symbols. We could go on like this all day.
2. 3y + 2x − 45(6 − x)2
1. x = 2
2. 4x = 20
3. 3x + 10y − 12 = 0
Notice that expressions do not have equal signs, while equations must have equal signs. An equation is a statement that two expressions have the same value, and the equal sign is a necessity. "Equation" even starts with the same letters as "equal," which should be a good clue.
In order to be an expression or equation, the string of symbols must make sense. Catch you drift our?
1. xy − + − 34
2. + + +
3. x 2 ÷ ( ( (
The pieces of an expression separated by + and − signs are called terms. A term will be positive if it follows a + sign and negative if it follows a − sign. When we break an expression into terms, we need to be careful to keep the negative signs in front of terms that are being subtracted. We also need to be careful not to get our thumb in the way when we're striking it with our ball-peen hammer.
1. In the expression 3x + 2, the terms are 3x and 2.
2. In the expression 3x − 2, the terms are 3x and -2. That's because 3x − 2 means 3x + (-2). Don't forget that there's actually a plus sign in front of positive numbers, even if we can't always see them. We knew we never should have given them that cloak of invisibility.
3. The terms in the expression are 5x, 3y2, and .
Expressions that are multiplied together are called factors. You 'll need to come to grips with this information—it's just one of the factors of life.
1. In the expression 3x, 3 and x are both factors.
2. In the expression (3 − x)(2y2 + 9), (3 − x) and (2y2 + 9) are both factors.
3. 5, x, and y are all factors of 5xy.
Two expressions are said to be equivalent if they produce the same number for any possible value of the variable. We'll bet you want to see some examples of this stuff in action. Don't ask how we knew. We had a feeling.
The expressions x and x + 0 are equivalent, since for any value of x, x and x + 0 are the same thing. Sorry, zero, but you're pretty worthless.
One way of expressing the fact that these two expressions are equivalent is to write an equation: x = x + 0. Any time you have an equation, the two expressions on either side of the equal sign are equivalent. That's kinda the equal sign's whole deal. It doesn't appreciate it when you write something like 1 + 1 = 3. Drives it batty.
The expressions x2 and (-x)2 are equivalent, since for any value of x we'll get the same value out of either expression. If x is 2, we find that 22 = 4 from the first expression and (-2)2 = 4 from the second expression. If x is 34,792, we get...oh, well, you probably got it from the first example. No need to wear ourselves out.
The goal is to create a toolbox of ways to modify an expression to an equivalent expression. We're not, of course, talking about a literal toolbox. You'll never receive an emergency call to hop over and fix an overflowing toilet with your trusty ratcheting box variable. Instead, these moves are the rules we have for working with algebraic expressions. Once we finish developing this (virtual) toolbox, we'll use our tools to solve equations and more. Okay, maybe the occasional leaky faucet, but that's where we draw the line.
There are several different things we can do to rearrange expressions. We've divided them into the following list of allowable moves:
5. Combining like terms
6. Getting rid of parentheses
Use any moves that aren't in the "toolbox," and you might draw a penalty flag. Incorrect Rearrangement of Expressions: 15 yards and loss of down.
Using combos of those moves together to rearrange an expression into an equivalent expression is called simplification.
This rule for rearrangement is almost so obvious it isn't even worth mentioning. But we're suckers for overkill, so here goes. The commutative property says that the order in which we add or multiply numbers doesn't matter.
x + y = y + x
xy = yx
This move, changing the order of our terms when we're adding or multiplying, is one of the most basic ways we can rearrange an expression. Note that we don't say you can also do this move with subtraction or division. That's because you can't. Addition and multiplication are a little more go-with-the-flow; subtraction and division aren't quite as easygoing and could probably stand to chillax a bit.
Consider the expression -4y2x + x3. We can use the commutative property of addition to rewrite the whole thing as x3 – 4y2x.
We can also use the commutative property of multiplication to rewrite -4y2x as -4xy2.
Putting it together, we can rearrange -4y2x + x3 using commutativity (and yes, we did make that word up) to get x3 – 4xy2. We're left with an expression that doesn't feature any fewer terms than the original, but at least it doesn't start out with a negative sign. That really chaps our patootie.
Another rule for rearrangement shows up when we add or multiply more than two terms. If you ever have occasion to introduce these terms to anyone in a business setting, you can make yourself sound über-professional by referring to them as your "associates." Hence the name: the associative property. Here's what it looks like for addition:
(x + y) + z = x + (y + z)
And it also works for multiplication:
(xy)z = x(yz)
This is another one of these straightforward rules for rearrangement that we use all the time. It's a fine line between obvious and oblivious, and we wanted to make sure.
Say you've got nine ladies dancing, eight maids a-milking, seven swans a-swimming, six geese a-laying, five golden rings, four calling birds, three French hens, two turtle doves, and a partridge in a pear tree, and you want to know how many total things that is. Not to mention where in the world you're gonna keep them all.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
Now we can rearrange all those terms and group together the numbers are straightforward to add. We rewrite using commutativity and associativity to get:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 =
(1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5 =
10 + 10 + 10 + 10 + 5 = 45
That's 45 total Christmas-y things. You must be getting into the holiday spirit. Even if the roof of your car is covered in French hen droppings.
Suppose we have 3 baskets, each holding 2 apples and 4 oranges.
This is the same number of apples and oranges as if we had a bag with 6 apples and a bag with 12 oranges. Except that we now mysteriously no longer have our three baskets, which were handmade in Santa Fe and actually hold quite a bit of sentimental value for us. That's a shame.
Regardless of how we package them, the number of fruits remains the same. Not that it takes away the sting of having had our wicker receptacles stolen from right under our noses.
This is an example of the distributive property, which basically says that it doesn't matter how we "package" numbers when performing multiplication. To write the distributive property in symbols, we say that if a, b, and c are real numbers, then:
a(b + c) = ab + ac
When we go from the left side to the right side of this equation, we say we're "distributing a over the quantity (b + c)." We may not say that aloud often, but we certainly won't hesitate to type it. In fact, we just did.
Multiply 3x(x + 2).
Now the thing we're distributing is 3x, rather than a plain old number all by its lonesome. That's okay, because the distributive property still works.
3x(x + 2) =
3x(x) + 3x(2)
Remember how exponent notation works? If not, get a refresher here. If we distribute something that has a variable over a quantity in parentheses that also contains that variable, we use exponent notation to keep things tidy. Probably couldn't hurt to also spray it down with a few squirts of Glass Plus. Let's use exponents to finish up.
3x(x) + 3x(2) =
3x2 + 6x
Expand -2(a + b).
Be careful: When the value you're distributing has a negative sign, make sure you distribute the negative sign over everything in the parentheses. Your parents may have told you to stop spreading your negativity, but ignore them for now.
-2(a + b) = -2a – 2b
Ah, much better.
Expand -(c + d).
Having a negative sign by itself outside the parentheses is the same as having -1 outside the parentheses. The 1 is there; it's just hiding. Did you check under the bed? That's totally its favorite spot. To distribute the negative sign, you would simply multiply each term inside the parentheses by -1.
-(c + d) = -c – d
What's the expanded version of -(2a – 5b – 6 + 11c)?
Just multiply every stinkin' term inside those parentheses by -1.
-(2a – 5b – 6 + 11c) = -2a + 5b + 6 – 11c
By the way, since multiplication is commutative, the distributive property also works if we write the multiplication the other way around:
(b + c)a = ba + ca.
Use the distributive property to multiply (4x – y)(-3).
This is the same thing as -3(4x – y), so just multiply -3 by both terms and voila:
(4x – y)(-3) =
4x(-3) – y(-3) =
-12x + 3y
Use the distributive property to multiply (4 – x)(-1).
Same old, same old. Tack a -1 onto both terms.
(4 – x)(-1) =
4(-1) – x(-1) =
-4 + x
Ready to really get down to business? The distributive property still works even if the expression in parentheses has more than two terms. It's totally a team player.
What's the expanded version of 4(x + y + z)?
This one's not too bad. Slap a 4 onto each variable and we're done.
4(x + y + z) = 4x + 4y + 4z
The distributive property also works when we're multiplying expressions where both factors have multiple terms. So if you're a tennis player, it's like playing straight doubles rather than Canadian doubles. Or triples. Okay, the analogy sort of falls apart at this point. Ignore us and take a look at one more example.
Expand (3 + x)(y – 4).
Okay, for this one we'll use the distributive property twice. It's double-distributing time. Basically, we want to keep on distributin' until the day is done. Or at least until there's nothing left to distribute.
First we separate the 3 and the x in the first factor.
(3 + x)(y – 4) =
3(y – 4) + x(y – 4)
Then we distribute both terms separately like normal.
3(y – 4) + x(y – 4) =
3y – 12 + xy – 4x
Man, that's a lot of stuff to keep track of at once. Think how much easier it would be if you could shower, brush your teeth, eat breakfast, and get dressed all at the same time. What a life-saver that would be! Especially on mornings that your alarm didn't go off...
We can work the distributive property in reverse—we just need to check our rear view mirror first for small children.
When we rewrite ab + ac as a(b + c), what we're actually doing is factoring. A factor in this case is one of two or more expressions multiplied together. Factoring an expression means breaking the expression down into bits we can multiply together to find the original expression. Why would we want to break something down and then multiply it back together to get what we started with in the first place? Oh, who knows. Those crazy mathematicians have a lot of time on their hands. It actually will come in handy, trust us.
Factoring expressions is pretty similar to factoring numbers. Click here for a refresher.
Factor the expression 3x2 – 27xy. Check to see that your answer is correct
Since each term of the expression has a 3x in it (okay, true, the number 27 doesn't have a 3 in it, but the value 27 does), we can factor out 3x:
3x2 – 27xy =
3x(x – 9y)
We can check that our answer is correct by using the distributive property to multiply out 3x(x – 9y), making sure we get the original expression 3x2 – 27xy. We do, and all of the Whos down in Whoville rejoice.
The value 3x in the example above is called a common factor, since it's a factor that both terms have in common. If these two ever find themselves at an uncomfortable office function, at least they'll have something to talk about.
When we factor an expression, we want to pull out the greatest common factor. The greatest common factor is a factor that leaves us with no more factoring left to do; it's the finishing move. That would be great, because as much as we love factoring and would like nothing more than to keep on factoring from now until the dawn of the new year, it's almost our bedtime. Let's find ourselves a GCF and call this one a night.
Factor the expression 45x – 9y + 99z.
Right off the bat, we can tell that 3 is a common factor. So let's pull a 3 out of each term.
45x – 9y + 99z =
3(15x – 3y + 33z)
Wait a sec. We can factor this expression even further because all of the terms in parentheses still have a common factor, and 3 isn't the greatest common factor. Although it's still great, in its own way. Really, really great. As great as you can be without being the greatest. This is us desperately trying to save face.
Instead, let's be greedy and pull out a 9 from the original expression.
45x – 9y + 99z =
9(5x – y + 11z)
The terms in parentheses have nothing else in common to factor out, and 9 was the greatest common factor. Not that that makes 9 superior or better than 3 in any way; it's just that...well, 3 is simply...oy. Insert foot into mouth.
To find the greatest common factor for an expression, look carefully at all of its terms. The number part of the greatest common factor will be the largest number that divides the number parts of all the terms. The variable part of a greatest common factor can be figured out one variable at a time. Much easier. Both to do and to explain.
For each variable, find the term with the fewest copies. Use that number of copies (powers) of the variable. Finally, multiply together the number part and each variable part. This step will get us to the greatest common factor.
Sometimes we have a choice of factorizations, depending on where we put the negative signs. We'll show you what we mean; grab a bunch of negative signs and follow us...
Factor the expression -50x + 4y in two different ways.
First way: factor out 2 from both terms.
-50x + 4y =
2(-25x + 2y)
Second way: factor out -2 from both terms instead.
-50x + 4y =
-2(25x – 2y)
You can double-check both of 'em with the distributive property.
These factorizations are both correct. Neither one is more correct, so let's not get all in a tizzy. Which one you use is merely a matter of personal preference. Or maybe a matter of your teacher's preference, if your teacher asks you to do these problems a certain way. If they do, don't fight them on it. They're bigger than you. Or at least they were a few years ago.
Be Careful: Always check your answers to factorization problems. When you multiply factors together, you should find the original expression. This step is especially important when negative signs are involved, because they can be a tad tricky. In fact, you probably shouldn't trust them with your social security number. Especially if your social has any negatives in it.