Algebraic Expressions

Objective:

 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

What are 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, 3x2, 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,«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»5«/mn»«mn»2«/mn»«/mfrac»«mo»,«/mo»«mo»-«/mo»«mfrac»«mn»7«/mn»«mn»2«/mn»«/mfrac»«/math»   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. 

Number line and an Expression

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.

 Terms, Factors and Coefficients

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.
 

Monomials, Binomials and Polynomials

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:    4x2, 3xy, –7z, 5xy2, 10y, –9, 82mnp, etc.
  • Examples of binomials:      a + b, 4l + 5m, a + 4, 5 –3xy, z2 – 4y2, etc.
  • Examples of trinomials:      a + b + c, 2x + 3y – 5, x2y – xy2 + y2, etc.
  • Examples of polynomials:  a + b + c + d, 3xy, 7xyz – 10, 2x + 3y + 7z, etc.

Like and Unlike Terms

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.

Addition and Subtraction of Algebraic Expressions

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

 For example, to add 7x2 – 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 5x2 – 4y2 + 6y – 3 from 7x2 – 4xy + 8y2 + 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 – 4y2 is the same as adding 4y2 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.

Summary

  • 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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