In this article, we will discuss a part of mathematics that involves variables. We will discuss algebra, algebraic expressions, equations and inequalities.
Contents
Introduction to Algebra
Algebra in very simple terms is part of mathematics that involves variables.
A variable is a letter that represents either a specific number or all numbers.
In algebraic expressions, the variable represents either all numbers or all numbers with a few specific exceptions.
In algebraic equations, the variable represents one or more initially unknown values, and the goal is to solve for those specific values.
Now that we know what a variable is, let’s define some additional terms as used in algebra.
Terminologies in Algebra
Constant
A constant is a number or a symbol such as π that doesn’t change in value.
Term
A term is a product of constants and variables including powers of variables e.g 2, x, 2x2, z4, x2y3z4 etc.
Coefficient
A coefficient is the constant factor of a term e.g in 3x2, 3 is the coefficient.
When no constant is written e.g in x2, the coefficient is 1.
Expression
An expression is a collection of one or more terms joined by addition or subtraction e.g.
x+y, x2-y2, 1+x3+x6, x2+2xy+y2 etc.
Expressions don’t have equal signs.
Monomial
A monomial is an expression with exactly one term e.g.
2x2, 12, z, x2y3 etc.
Binomial
A binomial is an expression with exactly two terms e.g.
x+5, x2+y2, a+b etc.
Trinomial
A trinomial is an expression with exactly three terms e.g x2+2xy+y2.
Polynomial
A polynomial is an expression with any number of terms involving only one variable.
Equations
An equation is simply an expression with equal sign e.g x2+2xy+y2 = 0.
Linear
A term with a single power of a variable (that is no explicit exponent).
Quadratic
A quadratic is a term with the square of a single variable.
Sometimes the words linear and quadratic can describe individual terms, but they can also describe entire expressions involving a single variable.
In a linear expression, the highest power of the variable is 1.
In a quadratic expression, the highest power of the variable is 2.
Cubic
A term with the cube of a single variable.
Algebraic Expressions
We can simplify algebraic expressions by adding or subtracting like terms e.g (x3-3x2) + 2x2+5x3 = x3+5x3-3x2+2x2 = 6x3-x2.
When adding expression with parentheses, we can simply remove the parentheses and perform the addition directly e.g 2x+(4x+5y) = 2x+4x+5y = 6x+5y but when subtracting an expression in parentheses, we have to change each term to its opposite sign when we remove the parentheses e.g 2x-(4x+5y) = 2x-4x-5y = -2x-5y.
The FOIL Method
Let us now discuss multiplication of two binomial expressions each of which involves addition or subtraction in the form (a+b)(c+d).
We can use the distributive law in this case but there is a very convenient shortcut summarized by the mnemonic FOIL.
F = First (a+b)(c+d) [a*c]
O = Outer (a+b)(c+d) [a*d]
I = Inner (a+b)(c+d) [b*c]
L = Last (a+b)(c+d) [b*d]
The product of the binomials is the sum of those four individual products.
Let us look at an example (2x+y)(x+2y):
F = First 2x*x
O = Outer 2x*2y
I = Inner y*x
L = Last y*2y
(2x+y)(x+2y) = 2x2 + 4xy + xy + 2y2 = 2x2 + 5xy + 2y2
