In this article, we will discuss a part of mathematics that involves variables. We will discuss algebra, algebraic expressions, equations and inequalities.
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
A constant is a number or a symbol such as π that doesn’t change in value.
A term is a product of constants and variables including powers of variables e.g 2, x, 2x2, z4, x2y3z4 etc.
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.
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.
A monomial is an expression with exactly one term e.g.
2x2, 12, z, x2y3 etc.
A binomial is an expression with exactly two terms e.g.
x+5, x2+y2, a+b etc.
A trinomial is an expression with exactly three terms e.g x2+2xy+y2.
A polynomial is an expression with any number of terms involving only one variable.
An equation is simply an expression with equal sign e.g x2+2xy+y2 = 0.
A term with a single power of a variable (that is no explicit exponent).
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.
A term with the cube of a single variable.
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