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Polynomials: Combining "Like Terms" (page 2 of 2) Sections: Polynomial basics, Combining "like terms" Probably the most common thing you will be doing with polynomials is "combining like terms". This is the process of adding together whatever terms you can, but not overdoing it by adding together terms that can't actually be combined. Terms can be combined if they have the exact same variable part. Here is a rundown of what's what:
Once you have determined that two terms are indeed "like" terms and can therefore be combined, you deal with them in a manner similar to what you did in grammar school. When you were first learning to add, you would do "five apples and six apples is eleven apples". You have since learned that, as they say, "you can't add apples and oranges". That is, "five apples and six oranges" is just a big pile of fruit; it isn't something like "eleven applanges". Combining like terms works much the same way. Here are some examples:
These are like terms since they have the same variable part, so combine the terms: three x's and four x's makes seven x's: Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved 3x + 4x = 7x
It is often best to group like terms together first, and then simplify: 2x2
+ 3x – 4 – x2 + x + 9
In the second line, many students find it helpful to write in the understood coefficient of 1 in front of variable expressions with no written coefficient, as is shown in red below: (2x2
– x2) + (3x + x) + (–4 + 9)
It is not required that the understood 1 be written in when simplifying expressions like this, but many students find this technique to be very helpful. Whatever method helps you consistently complete the simplification is the method you should use.
10x3
– 14x2 + 3x – 4x3 + 4x
– 6
The first thing I need to do is take the negative through the parentheses: 25 – (x
+ 3 – x2)
If it helps you to keep track of the negative sign, put the understood 1 in front of the parentheses: 25 – (x
+ 3 – x2)
While the first format (without the 1's being written in) is the more "standard" format, either format should be acceptable (check with your instructor). You should use the format that works most successfully for you.
This is the kind of problem that us math teachers love to put on tests (yes, we're cruel people), so you should expect to need to be able to do this. This is just an order of operations problem with a variable in it. Work carefully from the inside out, and you should be fine: x
+ 2(x – [3x – 8] + 3)
Work from the inside out: [(6x
– 8) – 2x] – [(12x – 7) – (4x – 5)]
–4y
– [3x + (3y – 2x + {2y – 7} ) - 4x
+ 5]
If you think you need more practice with this last type of problem (with all the brackets and parentheses, review the "Simplifying with Parentheses" lesson.) Don't get careless and confuse multiplication and addition. This may sound like a silly thing to say, but it is the most commonly-made mistake (after order of operations): (x)(x) = x2 (multiplication) x + x = 2x (addition) " x2 " DOES NOT EQUAL " 2x " So, if you have something like x3 + x2, DO NOT say that this somehow equals something like x5 or 5x. If you have something like 2x + x, DO NOT say that this somehow equals something like 2x2. << Previous Top | 1 | 2 | Return to Index
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