Algebraic Expressions

Definition
An algebraic expression is an expression built up from constants, variables, and a finite number of algebraic operations (addition, subtraction, multiplication, division and exponentiation(raised to an index) by an exponent(index) that is a rational number).[1] For example, expressions a and b are algebraic expressions.

Parts of an Algebraic Expression / Polynomial

 * 1) G3.png exponent (index/power).
 * 2) Coefficient of x2.
 * 3) An algebraic term (consists of coefficient, variable, both can have exponents)
 * 4) Arithmetic Operation
 * 5) Constant (numerical item)
 * 6) x and y - Variables/Unknowns

Algebraic Identities
Algebraic Identities are useful tools in the expansion and factorisation processes involving algebraic expressions. Three of the most common identities are listed below:
 * (a+b)2 = a2+2ab+b2,
 * (a-b)2 = a2-2ab+b2,

Binomial Theorem
It is possible to expand expressions in the form (x+y)n into a sum of terms of the form axbyc where b+c=n.



The coefficients of axbyc (or simply a), is called the binomial coefficient. The above polynomial expansion by the binomial theorem have the coefficients, 1, 4, 6, 4, 1. Which resembles the 4th (or simply the value of n) row of the Pascal's Triangle.



Similarly, the Pascal's Triangle can be used to expand polynomials in the form (x+y)n where the value of n indicates which row of binomial coefficients is used in the expanded expression.

Binomial Formula
The binomial formula, with respect to the binomial coefficient a, is used to represent the value of a inside the Pascal's Triangle. In general, n in the formula is the value of n inside the given polynomial of (x+y)n. While k is the position of the binomial coefficient with reference to Pascal's Triangle.

For a detailed explanation, given the polynomial (x+y)5, we can rewrite the binomial theorem in the form,



Alternatively, using the summation symbol, we write,



Using the derived form of binomial theorem, we then do the expansion process with reference to Pascal's Triangle. We get the expanded polynomial, x5+5x4y+10x3y2+10x2y3+5xy4+y5, which resembles k in the 5th row of Pascal's Triangle.