3.1-3.2 Expressions Notes
Terms
A term is the product of a number and one or more variables. A term may also be a single number, with no variable.
The expression14 + 3y2- 15zphas three terms:14,3y2, and-15zp.
The expression7z+ 12 + 2 +zhas four terms:7z, 12, 2,z.
The coefficient is the number that is multiplied by the variable(s) in a single term. In the expression14 + 3y2- 15zp,y2has a coefficient of3andzphas a coefficient of-15. A term with no coefficient, likez, has an implied coefficient of1.
Terms that do not contain variables are called constants. In the above expressions,14,12, and2are constants.
Combining Like Terms
Like terms are terms that contain the exact same variables raised to the same exponents. For example,15yzand22yzare like terms, but15yz2and22yzare not. Likewise,12w2yzand-5w2yzare like terms, but12w2yzand-5w2zare not.12and-6are like terms, because they are both constant terms.
To combine like terms, group them together in the equation, putting the terms with the highest exponents on the left. We can do this because additioncommutes.Next, group the coefficients of like terms together , all multiplied by the variable(s) in those terms. Finally, add the coefficients of the like terms (or subtract them if they are negative). Here are some examples:
Example 1: Simplify4y+ 15 - 2y+ 5y2+ 12 - 6.
- 5y2+ 4y- 2y+ 15 + 12 - 6(note that this expression and the original expression both contain 6 terms)
- 5y2+ (4 - 2)y+ (15 + 12 - 6)
- 5y2+ 2y+ 21
Example 2: Simplify11wz+ 6w2-10wz+wz- 6 - 2w+ 3w2.
- 6w2+3w2+ 11wz- 10wz+wz- 2w- 6(note that this expression and the original expression both contain 7 terms)
- (6 + 3)w2+ (11 - 10 + 1)wz- 2w- 6
- 9w2+ 2wz- 2w- 6
Example 3: Simplify-12w2+6w- 7w3+4w2+8w2- 2w+ 5 + 9w.
- -7w3-12w2+4w2+8w2+ 6w- 2w+ 9w+ 5(8 terms)
- -7w3+ (- 12 + 4 + 8)w2+ (6 - 2 + 9)w+ 5
- -7w3+0w2+13w+ 5 = - 7w3+ 13w+ 5
When we combine like terms, we convert the expression to simplified form. The expression is simplified form is equivalent to the original expression.