Multiplication of Algebraic Expression

Introduction on Multiplication of Algebraic Expression

Algebraic expressions explain a set of operations that should be done following a specific set of orders. Such expressions consist of an amalgamation of integers, variables, exponents, and constants. When these expressions undergo the mathematical operation of multiplication, then the process is called the multiplication of algebraic expression. Two different expressions that give the same answer are called equivalent expressions. Some other properties like distributive and commutative property of addition will come in handy while doing multiplying polynomials. We will discuss the multiplication of algebraic expressions later, but first, we need to understand some terms used in algebra.

Algebraic Terms

The parts or terms of an algebraic expression consist of the following:

  • Integers: An integer is any positive, negative, or zero number, but it has to be whole numbers (not a fraction or decimal).

  • Variables: When alphabets or symbols are used in a mathematical problem to represent a specific value, then they are called variables.

  • Exponents: The exponent in mathematical expressions is the number that represents the number of times the quantity has been multiplied by itself. It is also called the power or indices of the quantity. For example: In \[m^{3}\], the number 3 is an exponent, and the term represents that ‘m’ is raised to the power of 3, and \[m^{3}\] is equal to (m*m*m). 

  • Constant: The terms in an algebraic expression that comprises only numbers (no variables) are called constants, for example, in the expression 4x - 5y + 7, the constant term is 7. 

Polynomial Expression

The algebraic expression involving one or more terms, which comprise of variables, coefficients, exponents, and constants and combined by using mathematical operations like addition, subtraction, multiplication, and division, is called a polynomial. 

An expression is only considered to be polynomial in the absence of following elements- fraction power of the variable, negative exponents of a variable, square-roots of variables, and variables in the denominator. 

The mentionable types of polynomial expressions are Monomial, Binomial, and Trinomial, and their names are such because the expressions involve one, two, and three terms, respectively.

  1. Monomial expression: The algebraic expression involves only one term formed by the combination of integer and variable, for example: 3x, 4a, 5b, 3abx, etc. where x, a, b are the variables and 3, 4, 5 are called the coefficients.

  2. Binomial expression: The polynomial expression involves two terms, for example, 2x - 1, xy - 5z², etc.

  3. Trinomial expression: The algebraic expression that involves three terms, for example, 5x + 3y - 2, 7y² + 9y + 11, etc.


Multiplying Algebraic Expressions:

While doing multiplication of algebraic expressions, one should know about the proper operations of addition, subtraction, multiplication, and division of numeric values and variables.

Some rules that must be remembered while multiplying algebraic expression are:

  • The product two factors with the same signs will be positive, and the outcome of multiplying two terms with two, unlike signs, will be negative.

  • If x is variable and a, b are positive integers then, (xa * xb) = x(m +n) 


Multiplication of Monomial by Monomial :

The multiplication of two or more monomial expressions or expressions with one term means finding the product of all the expressions involved. While multiplication of monomials by monomial expressions the rule or equation that applies is mentioned below.

The product of monomials = (product of their coefficients)*(product of the variables).

Multiplication of Algebraic Expressions Examples:

  1. a*a = a²

  2. 2a*2b = (2*2)*(a*b) = 4ab

  3. 6ab*3x = (6*3)*(ab*x) = 18abx

  4. 5xy * 4x² * 2x³= (5*4*2)*{x(1+2+3) * y} = 40yx⁶ 

  5. 3x * (-5a²)(2b³) = (3 * 2) * (-5) * (x * a² * b³) = -15 x a² b³ 

Multiplying Monomials and Polynomials:

The rule that applies to the multiplication of monomials and polynomials is the distributive law. 

The law shows that each term of the polynomial should be individually multiplied by the monomial expression, x*(y +z) = (x*y) +(xz) = xy +xz and x*(y –z) = xy - xz

Multiplication of Monomials and Polynomial Examples:

  1. a*(a +b) = a² +ab

  2. 4xy(3xy) = 12(xy)²

  3. (-ab)*(a –b +c) = -a²b +ab² –abc

  4. 3xyz(x +2y -3) = 3yzx² +6xzy² -3xyz

  5. 5xy²*(3x+7y) = (5*3)*x(1 +1)*y² +(5*7)*x*y(2 +1) = 15x²y² + 35xy³

FAQ (Frequently Asked Questions)

What are the Practical Uses of Multiplication of Algebraic Expressions?

Teachers often have to face questions on the practical applications of algebraic expressions. To answer those giving an example would be best. 

A valid example would be the area measurement of rectangular space say of length (x +y) and breadth x. The area of that rectangle will be (xy +x²). This was calculated by multiplying algebraic expression [x*(x +y)]. Say, if the length and breadth are binomial in nature, then one will calculate the area by multiplying both the expressions. 

Any unknown value for a given circumstance can be taken as the variable, and an algebraic equation can be constructed by the right placing of known and unknown values. From that equation, the value of the unknown quantity can be calculated. To solve an equation, you must know the operations used in the algebraic expressions for learning go through the given multiplication of algebraic expression examples.

What are some of the Common Mistakes made while doing Multiplication of Algebraic Expression?

The common errors that students make during multiplying algebraic expressions are:

  • While multiplying monomials and polynomials, students often forget to multiply the monomial term with all the terms of the polynomial. For example, while calculating 3(a +b), one might wrongly solve it as (3a +b), while the right answer will be (3a +3b).

  • Another common mistake that students make is while multiplying with negative terms. An easy way to calculate the sign before a term is to check whether the same sign is before both the terms or are the signs before the two terms are different (one negative and the other positive).

  • A mistake that students often make while multiplying two or more polynomial expressions is getting the number of terms wrong in the product expression. An easy way to reduce mistakes and to understand the number of terms one should get in after multiplying the polynomials is by calculating the product of the number of all the terms involved.