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Algebra - Basic Definitions

It may help you to read Introduction to Algebra first

What is an Equation

An equation says that two things are equal. It will have an equals sign "=" like this:

x + 2 = 6

That equations says: what is on the left (x + 2) is equal to what is on the right (6)

So an equation is like a statement "this equals that"

Parts of an Equation

So that people can discuss equations, there are names for different parts (better than saying "that thingy there"!)

Here we have an equation that says 4x-7 equals 5, and all its parts:

A Variable is a symbol for a number we don't know yet. It is usually a letter like x or y.

A number on its own is called a Constant.

A Coefficient is a number used to multiply a variable (4x means 4 times x, so 4 is a coefficient)

An Operator is a symbol (such as +, ×, etc) that represents an operation (ie you want to do something with the values).

   

A Term is either a single number or a variable, or numbers and variables multiplied together.

An Expression is a group of terms (the terms are separated by + or - signs)

So, now we can say things like "that expression has only two terms", or "the second term is a constant", or even "are you sure the coefficient is really 4?"

Exponent!

8 to the Power 2

The exponent (such as the 2 in x2) says how many times to use the value in a multiplication.

Examples:

82 = 8 × 8 = 64

y3 = y × y × y

y2z = y × y × z

Exponents make it easier to write and use many multiplications

Example: y4z2 is easier than y × y × y × y × z × z, or even yyyyzz

Polynomial

Example of a Polynomial: 3x2 + x - 2

A polynomial can have constants, variables and the exponents 0,1,2,3,...

And they can be combined using addition, subtraction and multiplication, ... but not division!

polynomial

Monomial, Binomial, Trinomial

There are special names for polynomials with 1, 2 or 3 terms:

monomial, binomial, trinomial


Like Terms

"Like terms" are terms whose variables (and their exponents such as the 2 in x2) are the same.

In other words, terms that are "like" each other. (Note: the coefficients can be different)

Examples:

Terms Why are they "Like Terms"
arrow 7x x -2x because the variables are all x
arrow (1/3)xy2 -2xy2 6xy2 because the variables are all xy2

You can add like terms together to make one term:

Example: 7x + x = 8x

Unlike Terms

If they are not like terms, they are called "Unlike Terms":

Terms Why are they "Unlike Terms"
arrow -3xy -3y 12y2 these are all unlike terms
(xy, y and y2 are all different)