# Algebra

* Use of letters and symbols to represent values and their relations, especially for solving equations. This is also called "Elementary algebra". Historically, this was the meaning in pure mathematics too, like seen in "fundamental theorem of algebra", but not now. * In modern pure mathematics,

* a major branch of mathematics which studies relations and operations. It's sometimes called abstract algebra, or "modern algebra" to distinguish it from elementary algebra. * a mathematical structure as a "linear" ring, is also called "algebra," or sometimes "algebra over a field", to distinguish it from its generalizations.

A variable is a letter or symbol that takes place of a number in Algebra. Common symbols used are a, x, y, θ, and λ. The letters x and y are commonly used, but remember that any other symbols would work just as well. Variables are used in algebra as placeholders for unknown numbers. If you see "3 + x", don't panic! All this means is that we are adding a number who's value we don't yet know. Term: A term is a number or a variable or the product of a number and a variable(s). An expression is two or more terms, with operations between all terms. Polynomial: Mathematical sentence with "many terms" (literal English translation of polynomial). Terms are separated by either a plus (+) or a minus (-) sign. There will always be one more term than there are plus (+) or minus (-) signs. Also, the number of terms will (generally speaking) be one higher than the lead exponent. Base: The number directly preceding an exponent

EX: a2 -> a is the base

Exponent: The number (written in superscript) used to express how many times a base is multiplied by itself EX: a4 = a * a * a * a -> 4 is the exponent

EX: 43 = 4 * 4 * 4 = 64 -> 3 is the exponent

EX: A Quadratic function has a lead exponent of 2, but generally has three terms (ax2 + bx + c; lead exponent = 2, # of + (or -) signs = 2, # of terms = 3) A polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, and multiplication (where repeated multiplication of the same variable is conventionally denoted as exponentiation with a constant nonnegative integer exponent). For example, x2 + 2x − 3 is a polynomial in the single variable x. An important class of problems in algebra is factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials. The example polynomial above can be factored as (x − 1)(x + 3). A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.

Exponents are a simple way to represent repeated multiplication. For example a x a = a2. There are a few simple rules for exponents that help reduce very large problems to simple little ones. The rules are as follows: 1) The exponent of any number is always a one (1): a = a1

2) When we multiply the same base we add our exponenents: a3 x a2 = a3 + 2 = a5 3) When we divide the same base we subtract our exponents: a6 / a4 = a6 - 4 a2 4) When we raise a power to a power we multiply our exponents: (a2)3 = a2 * 3 = a6 5) When we raise a PRODUCT to a power we raise both parts of the product to the power: (ab)3 = a3b3 [NOTE: This ONLY works with multiplication and NOT addition: (a + b)3 ≠ a3 + b3] 6) When we raise a QUOTIENT to a power we raise both parts of the quotient to the power: (a/b)2 = a2 / b2 [NOTE: This ONLY works with division and NOT subtraction: (a - b)2 ≠ a2 - b2] There are two very important things you need to know when working with Zero Power or Negative Exponents. First, any number to the Zero Power always equals one. For example (-500 = 1) There is one number that CANNOT be raised to the Zero Power, 00 does not exist! When dealing with Negative...

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