Algebraic Expressions

Algebraic Expressions

A mathematical statement which is obtained by using mathematical fundamental operation between constants and variables with different powers is called an algebraic expression.

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For example:

-     The sum of 2 and x is 2 + x, where the constant 2 and variable x is connected by ‘+’ sign.

-     So, 2 + x is an algebraic expression.

-     Similarly, other algebraic expressions are 2xy, 3x + 4y, x – y² + 4 , etc.

Algebraic Terms

The parts of an algebraic expression separated by ‘+’ or ‘–’ signs are called algebraic terms.

For example:

-     3x + 2y is an algebraic expression, where 3x and 2y are two parts connected by ‘+’ sign. So, 3x and 2y are called the terms of the expression 2x + 3y.

Types of algebraic expression

Algebraic expressions are of different types. They are: Monomial, Binomial, Trinomial and multinomial expressions.

-     Monomial Expression: An algebraic expression which contains only one term is called a monomial expression. For example: 4x, 3x²y, -7xyz², etc.

-     Binomial Expression: An algebraic expression which contains two terms connected by ‘+’ or ‘–’ is called a binomial expression. For example: a + b, 2a² – 3abx², 3x – 4ay², etc.

-     Trinomial Expression: An algebraic expression which contains three terms connected by ‘+’ or ‘–’ is called a trinomial expression. For example: x + 2y – z, a² + 2ab + b², etc.

-     Multinomial Expression: An algebraic expression which contains four or more than four terms connected by ‘+’ or ‘–’ is called a multinomial expression. For example: 2x – 2y + z, x³ – a² + 2ab – b² + 7, etc.

Coefficient, base and power

-     Coefficient: A numerical or constant quantity placed before the variable and multiplying it is called the coefficient. For example: In an expression 3x, 3 is called the coefficient of x.

-     Base: An alphabet letter used in algebraic expression is called the base. For example: In an expressioin 3x², x is the base.

-     Power: The repeatation of the same variable for the required number of times in an algebraic expression is called power. For example: In an expressioin 3x², 2 is the power of x.

Like and Unlike Terms

-     Like Terms: The terms are like if the terms have the same base with same power of the variable. For example: 2a, -3a, 4a are like terms. x³, 2x³, -5x³, -3x³ are like terms. similarly 7xy, -3xy, 2xy are like terms.

-     Unlike Terms: The terms are unlike if the terms have the different variables or different powers of the base. For example: x, 2x², 3x²y, -y are unlike terms. x³, x⁴, x^-3, -x are unlike terms. Similarly xy, 2xz, 3x²y are unlike terms.

Values of Algebraic Expressions

When we substitute the value of a variable in an algebraic expression with a number, the value of the expression is obtained after use of mathematical fundamental operations, it is called the value of the expression.

For example: if x = 2 and y = 3, find the value of 2x – 3x²y.

Here, the given expression = 2x – 3x²y

                                       = 2 × 2 – 3 × 2² × 3

                                       = 4 – 3 × 4 × 3

                                       = 4 – 36

                                       = -32

Workout Examples

Example 1: Write the type of each expression.

a.     4x

b.    2a + 3b

c.     3x² – 4xy + 2y²

d.    9x³ – 10x² + 4x – 21

e.     3x³ – 2x² + 5x + 7xy - 8

Solution: Here,

   a.     4x -------> Monomial

   b.    2a + 3b ---------> Binomial

   c.     3x² – 4xy + 2y² ---------> Trinomial

   d.    9x³ – 10x² + 4x – 21 ----------> Multinomial

   e.     3x³ – 2x² + 5x + 7xy – 8 ----------> Multinomial

Example 2: Rewrite the following statements in algebraic expressions.

a.     The sum of twice x and 5.

b.    Three times the difference of x and y is less than 3a.

c.     5x is added to the product of b and 7.

d.    One fourth of x is added to 25

e.     7x is divided by 14 and added to product of c and 4.

Solution: Here,

   a.     2x + 5

   b.    3(x – y) < 3a

   c.     5x + 7b

   d.    x/4 +25

   e.     7x/14 +4c

Example 3: Write the terms having:

a.     Base = x, power = 2 and coefficient = 1

b.    Base = x, power = 3 and coefficient = -3

c.     Base = y, power = 5 and coefficient = 4b

d.    Base = z, power = 2 and coefficient = -a

Solution: Here,

   a.     x²

   b.    -3x³

   c.     4by⁵

   d.    –az²

Example 4: Identify the like and unlike terms of the following expressions:

a.     3x, -2x, 7x

b.    3a², 3a³, 4a

c.     2(x+y), 3(x+y)³, 9(x+y)²

Solution: Here,

             a.     Like terms

             b.    Unlike terms

             c.     Unlike terms

Example 5: If x = 4 and y = 3, find the value of 2xy + 3y²

Solution: Here,

                        x = 4

                        y = 3

                        ∴ 2xy + 3y² = 2 × 4 ×3 + 3 × 3²

                                            = 24 + 3 × 9

                                            = 24 + 27

                                            = 51

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