Glossary for Lesson 2: Expressions

Factors

The factors of a term or expression are numbers, variables, or combinations of numbers and variables that, when multiplied together, produce that term or expression.  For example, 2 and 3 are factors of 6, and -1, 5, x, and y are factors of -5xy

Commutative Property

For addition: The commutative property of addition states that changing the order in which numbers are added does not affect the final result.  For example, 2+3=3+2.  This is true no matter how many terms are being added together. For multiplication: The commutative property of multiplication states that changing the order in which numbers are multiplied does not affect the final result.  For example, 2\cdot3=3\cdot2.  This is true no matter how many terms are being multiplied together.

Combining Like Terms

Like terms are terms which have the same variables with the same powers.  7xy^2 and -\frac {3}{2}xy^2 are like terms, while 7xy and -\frac {3}{2}xy^2 are not.  Since the only thing that differs between like terms is the coefficient, they can be added together, and this is called combining like terms

Polynomial

An expression made up of variables and/or constants that are combined with addition, subtraction, and multiplication and have non-negative integer exponents.  Examples: -3x+4, 6y^4+7y^2-8y+13, 5xy^2z-11

Degree of a Polynomial

To find the degree of a polynomial, we first need to find the degree of each of its terms.  The degree of a term is the sum of the powers of all of the variables in that term.  So, for instance, 7x^2 has a degree of 2, and -4xy^2z has a degree of 4.  The degree of a polynomial is equal to the degree of its highest-degree term.  Examples: 6y^4+7y^2-8y+13 has a degree of 4 or, in other words, is a fourth degree polynomial.

Standard Form of a Polynomial

A polynomial is said to be in standard form when it is written with its highest-degree term first, with the rest of the terms written in order of descending degree.  For example, 6y^4+7y^2-8y+13 is written in standard form, whereas 7y^2-8y+13+6y^4 is not.

Exponent

Repeated multiplication can be written using exponential notation.  For example, 3\cdot3\cdot3\cdot3\cdot3=3^5.  Here, the number 3 is known as the base, and 5 is known as the exponent or the power.  The base thus refers to the quantity being multiplied by itself, and the exponent tells us how many times this multiplication happens.

Order of Operations

The order of operations establishes the order in which different mathematical procedures should be performed.  This is often expressed with the acronym PEMDAS (or perhaps BEDMAS, BIDMAS, or BODMAS).  PEMDAS, which stands for Parentheses Exponents Multiplication Division Addition Subtraction, reminds us that when simplifying expressions, we should start by simplifying all expressions inside parentheses, then apply exponents, then multiply or divide, and lastly add or subtract.
Subtraction and addition have the same level of priority, so when terms are being added or subtracted one after the other, we simply work from left to right.  For example, if we had 1-4+7 , we would simplify it as follows: 1-4+7=-3+7=4.
Multiplication and division have the same sort of relationship: they have the same level of priority as one another, so when terms are being multiplied or divided one after the other, we simply work from left to right.  For example, if we had 3\div 6\times 9 , we would simplify it as follows: 3\div 6\times 9=2\times 9=18. The order of operations gives us a set of general guidelines that we must apply carefully.