Algebraic Expressions

This topic gives an overview of;

- Expressions
- Number line and an Expression
- Terms, Factors and Coefficients
- Monomials, Binomials and Polynomials
- Like and Unlike Terms
- Addition and Subtraction of Algebraic Expressions

An expression having one or more variables is called an **algebraic expression**. An algebraic expression may or may not contain mathematical operators like the symbols of addition, subtraction and multiplication. Examples of expressions are: x + 3, 2y – 5, 3x^{2}, 4xy + 7 etc.

You can form many more expressions. Expressions are formed from **variables and constants**. The expression 2y – 5 is formed from the variable y and constants 2 and 5. The expression 4xy +7 is formed from variables **x **and** y** and constants 4 and7.

We know that, the value of y in the expression, 2y – 5, may be anything. It can be 2, 5, –3, 0, etc.; actually countless different values.

The value of an expression changes with the value chosen for the variables it contains. Thus as *y* takes on different values, the value of 2y – 5 goes on changing. When y = 2, 2y – 5 = 2(2) – 5 = –1; when y = 0, 2y – 5 = 2 × 0 –5 = –5, etc.

Consider the expression x + 5. Let us say the variable x has a position X on the number line;

X may be anywhere on the number line, but it is definite that the value of x + 5 is given by a point P, 5 units to the right of X. Similarly, the value of x – 4 will be 4 units to the left of X and so on.

What about the position of 4x and 4x + 5?

The position of 4x will be point C; the distance of C from the origin will be four times the distance of X from the origin. The position D of 4x + 5 will be 5 units to the right of C.

Take the expression 4x + 5. This expression is made up of two terms, 4x and 5. Terms are added to form** expressions**. Terms themselves can be formed as the** product of factors**. The term 4x is the product of its factors 4 and x. The term 5 is made up of just one factor, i.e., 5.

The expression 7xy – 5x has two terms 7xy and –5x. The term 7xy is a product of factors 7, x and y. The numerical factor of a term is called its **numerical coefficient** or **simply coefficient**. The coefficient in the term 7xy is 7 and the coefficient in the term –5x is –5.

Expression that contains only one term is called a **monomial**. Expression that contains two terms is called a **binomial**. An expression containing three terms is a **trinomial** and so on. In general, an expression containing, one or more terms with non-zero coefficient (with variables having non negative exponents) is called a **polynomial**. A polynomial may contain any number of terms, one or more than one.

- Examples of monomials: 4x
^{2}, 3xy, –7z, 5xy^{2}, 10y, –9, 82mnp, etc. - Examples of binomials: a + b, 4l + 5m, a + 4, 5 –3xy, z
^{2}– 4y^{2}, etc. - Examples of trinomials: a + b + c, 2x + 3y – 5, x
^{2}y – xy^{2}+ y^{2}, etc. - Examples of polynomials: a + b + c + d, 3xy, 7xyz – 10, 2x + 3y + 7z, etc.

Terms that have the same power of the same variable are called **like terms**. Terms that do not contain the same power of the same variable are called **unlike terms**.

In the earlier classes, we have also learnt how to add and subtract algebraic expressions.

For example, to add 7x^{2} – 4x + 5 and 9x – 10, we do

Observe how we do the addition. We write each expression to be added in a separate row. While doing so we write like terms one below the other, and add them, as shown. Thus 5 + (–10) = 5 –10 = –5. Similarly, – 4x + 9x = (– 4 + 9)x = 5x.

Let us take some more examples.

**Example 1:**** Add:** 7xy + 5yz – 3zx, 4yz + 9zx – 4y , –3xz + 5x – 2xy.

Solution: Writing the three expressions in separate rows, with like terms one below the other, we have

Thus, the sum of the expressions is 5xy + 9yz + 3zx + 5x – 4y. Note how the terms, – 4y in the second expression and 5x in the third expression, are carried over as they are, since they have no like terms in the other expressions.

Example 2: Subtract 5x^{2} – 4y^{2} + 6y – 3 from 7x^{2} – 4xy + 8y^{2} + 5x – 3y.

Subtraction of a number is the same as addition of its additive inverse. Thus subtracting –3 is the same as adding +3. Similarly, subtracting 6y is the same as adding – 6y; subtracting – 4y^{2} is the same as adding 4y^{2} and so on. The signs in the third row written below each term in the second row help us in knowing which operation has to be performed.

- Expressions are formed from variables and constants.
- Terms are added to form expressions. Terms themselves are formed as product of factors.
- Expressions that contain exactly one, two and three terms are called monomials, binomials and trinomials respectively.
- In general, any expression containing one or more terms with non-zero coefficients (and with variables having non-negative exponents) is called a polynomial.
- Like terms are formed from the same variables and the powers of these variables are the same, too. Coefficients of like terms need not be the same.
- While adding (or subtracting) polynomials, first look for like terms and add (or subtract) them; then handle the unlike terms.
- There are number of situations in which we need to multiply algebraic expressions: for example, in

finding area of a rectangle, the sides of which are given as expressions.

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