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 trying to add together terms that can't actually be combined. Terms can be combined ONLY IF they have the exact same variable part. Here is a rundown of what's what:

 4x and 3 NOT like terms The second term has no variable 4x and 3y NOT like terms The second term now has a variable, but it doesn't match the variable of the first term 4x and 3x2 NOT like terms The second term now has the same variable, but the degree is different 4x and 3x LIKE TERMS Now the variables match and the degrees match

Once you have determined that two terms are indeed "like" terms and can indeed therefore be combined, you can then 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.

• Simplify 3x + 4x

These are like terms since they have the same variable part, so I can combine the terms: three x's and four x's makes seven x's:   Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

3x + 4x = 7x

• Simplify 2x2 + 3x – 4 – x2 + x + 9

It is often best to group like terms together first, and then simplify:

2x2 + 3x – 4 – x2 + x + 9
=  (2x2x2) + (3x + x) + (–4 + 9)

=
x2 + 4x + 5

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:

(2x2x2) + (3x + x) + (–4 + 9)
= (2x2
1x2) + (3x + 1x) + (–4 + 9)
= 1x2 + 4x + 5

=
x2 + 4x + 5

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.

• Simplify 10x3 – 14x2 + 3x – 4x3 + 4x – 6
• 10x3 – 14x2 + 3x – 4x3 + 4x – 6
=  (10x3 – 4x3) + (–14x2) + (3x + 4x) – 6

=
6x3 – 14x2 + 7x – 6

Warning: When moving the terms around, remember that the terms' signs move with them. Don't mess yourself up by leaving orphaned "plus" and "minus" signs behind.

• Simplify 25 – (x + 3 – x2)

The first thing I need to do is take the negative through the parentheses:

25 – (x + 3 – x2)
=  25 – x – 3 + x2

=  x2x + 25 – 3

=
x2x + 22

If it helps you to keep track of the negative sign, put the understood 1 in front of the parentheses:

25 – (x + 3 – x2)
=  25 –
1(x + 3 – x2)
=  25 – 1x – 3 + 1x2

=  1x2 – 1x + 25 – 3

=  1x2 – 1x + 22

= x2 – 1x + 22

While the first format (without the 1's being written in) is the more "standard" format, either format should be acceptable (but check with your instructor). You should use the format that works most successfully for you.

• Simplify x + 2(x – [3x – 8] + 3)

Warning: 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. If I work carefully from the inside out, paying careful attention to my "minus" signs, then I should be fine:

x + 2(x – [3x – 8] + 3)
=  x + 2(x – 1[3x – 8] + 3)

=  x + 2(x – 3x + 8 + 3)

=  x + 2(–2x + 11)

=  x – 4x + 22

=
–3x + 22

• Simplify [(6x – 8) – 2x] – [(12x – 7) – (4x – 5)]

I'll work from the inside out:

[(6x – 8) – 2x] – [(12x – 7) – (4x – 5)]
=  [6x – 8 – 2x] – [12x – 7 – 4x + 5]

=  [4x – 8] – [8x – 2]

=  4x – 8 – 8x + 2

=
–4x – 6

• Simplify –4y – [3x + (3y – 2x + {2y – 7} ) – 4x + 5]
• –4y – [3x + (3y – 2x + {2y – 7} ) - 4x + 5]
=  –4y – [3x + (3y – 2x + 2y – 7) - 4x + 5]

=  –4y – [3x + (–2x + 5y – 7) – 4x + 5]
=  –4y – [3x – 2x + 5y – 7 – 4x + 5]

=  –4y – [3x – 2x – 4x + 5y – 7 + 5]

=  –4y – [–3x + 5y – 2]

=  –4y + 3x – 5y + 2

=  3x – 4y – 5y + 2

=
3x – 9y + 2

If you think you need more practice with this last type of problem (with all the brackets and the negatives and the parentheses, then review the "Simplifying with Parentheses" lesson.)

Warning: 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 messing up the order of operations):

(x)(x) = x2     (multiplication)

x + x = 2x     (addition)

" x2DOES NOT EQUAL  " 2x "

So if you have something like x3 + x2, DO NOT try to 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.

 Cite this article as: Stapel, Elizabeth. "Polynomials: Combining 'Like Terms'." Purplemath. Available from     http://www.purplemath.com/modules/polydefs2.htm. Accessed [Date] [Month] 2016

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