Terms & Combining Like Terms

Terms

Terms, what they are and the process of combining like terms.

TERM

Simple definition: Any number, variable or constant that is separated by an addition or subtraction symbol.

????Why????

There is a simple reason why this is so, but in order to get a better understanding of what a term is and why we do what we do, when it comes to addition/subtraction or multiplication/division we need to re-examine what a number is.

A number is:

• A symbol used to represent a quantity.

• The product of a series of factors.

• The quotient of a series of divisors.

• An efficient way to represent large quatitites.

• 
  

The concept that all numbers are "the answer" to a series of multiplications or divisions will be important in the understanding of terms.

Remember that most numbers are either prime or composite. Even though 1 & 0, is neither prime or composite, all other numbers greater than 1 can be written as a series of multiplied prime numbers.

Any Number Greater than 1!!!!

7 is a prime number   $7\times1=7$  

8 is a composite number  $2\times2\times2\times1=8$  Even though 8 is composite, it can be written as a series of prime numbers.

17 is a prime number  $17\times1=17$  

35 is a composite number  $5\times7\times1=35$  

All numbers are ultimately multiplied or divided by...

Multiplication & Division

These two operations can be thought of as number "glue". Everytime anything is mutliplied or divided together, the entire group of numbers or variables IS A TERM.

Another View

A term is a quantity that has a positive or negative value. When the there are more than one of these quatitites the term is then multiplied. The example on the right show red and blue with the coefficients 4, for red, and 2 for blue. The red and the blue are seperated by a +; therefore, there are only two terms on both sides of the equal sign.

Poll

In the equation below, 

 $3+2x-4=2\cdot2-3x$  

How many terms are there?

• 3

• 4

• 5

• 6

Correct Answer - 5

There are only 5 terms. One of the terms is made up of 2 numbers because they are multiplied.

Since, both 2's after the equal sign are multiplied together the two numbers are considered ONE term. 

 $3+2x-4=2\cdot2-3x$  

Combination of Like Terms

The combination of like terms is similar to the integer addition rules where you made a T-Chart and placed all the negative numbers on one side, and all of the positive numbers on the other.

What is different is that the common factor is a variable.

Common Factor - Quick Review

For Example,

 $8=2\cdot4$  
 $20=5\cdot4$  

The common  factor is 4, because it can be found in both sets of factors. 

Combining Like Terms - Continued

Combining like terms is finding terms with a common factor, and combining the coefficients. This common factor is usually a variable because if the terms are simply numbers, we apply integer rules.

For Example,

8+20=28

.......................................or.............................

2 x 4 + 5 x 4 = 28

Common Factor - 4, coefficients ( 2 & 5). Both coefficients are positive; therefore combining them would get us 7....and 7 x 4 = 28

With Variables.

12x + 3y + 6z - 2x + 7y - 8z

Identify terms with the same variables (common factor)

12x ... & ...- 2x

3y ... & ... 7y

6z ... & ... -8z

Use integer rules to COMBINE (remember the T-Chart Method)

10x

10y

-2z

The first 2 terms are positive, so your expression would be

10x + 10y -2z

!!!!!!!!!!!!!!!VERY IMPORTANT!!!!!!!!!!!!!

The addition and subtraction in front of the term is really a positive or a negative. So when you are combining like terms, you are using the rules for adding and subtracting negative numbers.

Multiple Choice

Which of the following is the correct combination of the the following terms:

 $3x\ +\ 5\ -x\ -2$  

• 4x + 7

• 8x - 2x

• 2x + 3 

• 5x - 5

Multiple Choice

Which of the following is the correct combination of the the following terms:

 $3x\ +\ 5\ +x\ +2$  

• 4x + 7

• 8x - 2x

• 2x + 3 

• 5x - 5

Multiple Choice

Does this expression have like terms?

 $5n+3n+4n$  

• Yes

• No

Yes

There were three terms in the expression and they all had the variable

n

so they were all like terms.

Constants

Constants are numbers, and since they all share the common factor -1-, they are like terms.

Multiple Choice

Which of the following is correct for the expression below?

 $7z+3-2z-9$  

•  $9z-12$  

•  $5z-12$  

•  $5z-6$  

•  $9z-6$