# Algebraic Expressions

Constants and Variables:

The height of a tree is $10$ meters long and it grows $1$cm 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.

Example:
Suppose a car covers a distance of $20$km 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 $t$ = 20$t$, $20$ 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 expression$3{x^2} + 4x + 6$; 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.