# 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.