Algebraic Expressions

Constants and Variables:

The height of a tree is 10 meters long and it grows 1cm in a year.
Then its height after one year = 10 meters 1 cm
Its height after 3 years = 10 meters 3 cm
Its height after 6 years = 10 meters 6 cm
Its height after x years = 10 meters x cm

Where x represents an unknown number. From the last line, we can find height of the tree after a certain number of years by taking x equal to that number. For example, we simply let x= 15, 25 and 55. Thus, the value of x depends on our choice. We can give x any value or number we want. In other words, the value of x is not fixed, it varies from one situation to the other. Therefore, we call x a variable whereas 10 is a fixed number whole value does not change. So 10 therefore is called a constant.

Suppose a car covers a distance of 20km in an hour.
The distance covered by a car in 2 hour = 20 x 2 km
The distance covered by a car in 3 hour = 20 x 3 km
The distance covered by a car in t hour = 20 x t km
In 20 x t = 20t, 20 is a constant and t is a variable, because t can be given any value. It is customary to denote a variable by either x,y,z or t and a constant by a,b,c,d,e or f.

Algebraic Expression:
An algebraic expression is obtained by combining constants and variables by means of the operations of addition, subtraction, multiplication and division.

Examples of algebraic expressions are:
2x,\,3x + 5,\,6{x^2} + 7x + 10,\,5x + 7y + 10
In the first three expressions x is the only variable, while in the fourth expression 5x + 7y + 10, x and y are the two variables. Likewise x + 2y + 3z + 5 is an algebraic expression with three variables, x,y and z.

Terms of an Algebraic Expression:
The signs “ ”, “-” separate the algebraic expression into its terms, for example:
(1) 2t has one term- 2t
(2) 3x + 2 has two terms- 3x and 2
(3) 5x - y + 7 has three terms- 5x,  - y and 7
(4) 3{x^2} + 5x - 10 has three terms- 3{x^2}, 5x and -10

Coefficients and Degree of an Algebraic Expression:
Consider the algebraic expressions 5{x^2},7{y^4} and 10{t^7}
In 5{x^2}, x is called the base, 2 is called the exponent of the base x, while 5 is called the coefficient of {x^2}. Exponents tell us how many times the base is multiplied by itself.

For example by {x^2} we mean x \times x , {y^3} = y \times y \times y and so on.

In 7{y^4}, y is the base, 4 is the exponent and 7 is the coefficient of {y^4}. In 10{t^7}, t is the base, 7 is the exponent and 10 is the constant before the variable {t^7} is the coefficient of {t^7}.

Now consider the algebraic expression3{x^2} + 4x + 6; the highest exponent of x occurring in the expression is 2, and we call it an algebraic expression of degree 2. Note that we will learn in later sections that a number whose exponent is zero is equal to one; thus we can also write 6 = 6{x^0} . Hence 3{x^2} + 4x + 6 = 3{x^2} + 4x + 6{x^0} has the three terms 3{x^2},4x and 6{x^0}, which have the coefficients 3, 4 and 6. The coefficients of an algebraic expression are the same as the coefficients of its terms. Thus, the coefficients of 3{x^2} + 4x + 6 are 3, 4 and 6. 6 is also called a constant term.