We are so used to working with numbers that when we suddenly see alphabets, we get confused. How can we add or subtract anything that has alphabets? What is the difference between “y multiplied by y” and “y added to y”? How can anyone divide alphabets?? This article helps to clear all the confusion related to Algebra in an easy and fun way.
The Variable and Coefficient
For starters, what is the alphabet called? It is a variable and the number that is associated with it, in Algebraic parlance , is called the coefficient.
Eg -: 5a, 5 is the coefficient and a is the variable
-3x, -3 is the coefficient and x is the variable
y, 1 is the coefficient(which is understood but not explicitly written) and y is the variable
Now, what does the term variable mean? As the word suggests, a variable can stand for any number. So, it is represented by an unknown, which is the alphabet. The coefficient is a constant whose numerical value cannot change.
Algebra is a branch of Mathematics that deals with constants and variable. Algebra as it grows more complicated, branches out into many more streams. The are so many applications to algebra , because as we go on further in Math , we realize that we need to start generalizing and cannot stick to constants all the time. We will have to start figuring out complex questions in which so many things are not known , but have to be found out. How do we depict them ? The unknowns are variables.
Term – Like and Unlike
A variable and a constant together are called term.
Eg-: x, xy , y3 , 2xy , y2
Terms can be of two types – like and unlike terms. Like terms have similar variables which have similar powers(x 3 – here x is raised to power 3 which meaning we will learn later) and unlike terms have different variables.
Eg -: x and x, xy and 12yx , y3 and 2y3
These are similar terms and mathematical operations like addition and subtraction can be performed on these terms.
x and y , y2 and y , xy and x are unlike terms and it is not possible to add them or subtract one from the other. Multiplication and division are possible. It’s like 5 chocolates and 6 mangoes cannot be added or subtracted from each other ! How can we determine which terms are like and which are unlike?
It is quite simple really. Take the two terms in question and ignore the coefficients , concentrate only on the variables. Check if each term has the SAME variables with the SAME powers. It does not matter if they are in different order. After all we know by commutative law of multiplication that 2 x 3 is the same as 3 x 2.
If the variables are the same, then the terms are like , if not they are unlike.
When we join the like and unlike terms together with mathematical operations of “+” and “-“ we get an Algebraic Expression.
Eg-: xy + y3
56a2b3 – 7ab5 + 21b3a2
xy + 23
These expressions can be classified into monomial, binomial and polynomial expressions based on the number of terms in the expression.
A monomial expression , as it’s name suggests “mono” has only one term in it’s expression. That term can be a variable , a constant or a variable with a coefficient. There is no + or – sign here.
Eg -: x
A binomial as it’s root “bi” suggests, has two unlike terms. If the two terms are like , then we can just add the terms together to get a monomial. So the terms have to be unlike.
Eg xy + xyz
12pq + 12 rp
10abc + abc is not binomial as it can be added to give 11abc
A trinomial has three unlike terms in the expression.
Eg 2ab2 + b3c + ab
An algebraic expression with more than 3 unlike terms is called a polynomial. Sometimes all expressions other than a monomial are called polynomials.
A monomial, binomial , trinomial or polynomial for that matter , should not contain negative powers. Powers of positive numbers alone is acceptable. Negative powers and variables cannot be used.
Eg a x, x/y(as this is equal to xy-1) , b2d-3 are all invalid and are not monomials.
An algebraic expression can be separated into terms at the + and – signs. A term can be broken into its factors. All the variables and the coefficient that make a up a term are called its factors.
Separation of an algebraic expression into its terms and factors can be done easily using a tree diagram
Take for example the algebraic expression xy3 + 12xy