Multiplication of Algebraic Expressions Explained Clearly

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 properties of addition will come in handy while multiplying polynomials. We will discuss the multiplication of algebraic expressions later, but first, we need to understand some terms used in algebra.

What are the Algebraic expressions in maths?

In mathematics, the equations or the expressions that consist of at least one variable and a constant can be considered as algebraic expressions. In these algebraic expressions, the various variables and constants are connected by different basic mathematical operations namely addition and subtraction.

For example, 2 x + 9 y, 3 y²,  and many more equations or expressions like this are an example of algebraic expressions. 

In the example of 2 x + 9 y, the mathematical numbers of 2 and 9 are considered as the constant, while the alphabetic letters of x and y are said to be the variables.

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 quantity. For example: In m³

• The number 3 is an exponent, and the term represents that ‘m’ is raised to the power of 3.

• m³

• 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 the 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 the 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 of 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) = 15x² y² + 35xy³

Multiplication of Two Binomials 

An expression is said to be a binomial when they are made up of two individual terms or monomials.

In this article, students get knowledge on the various aspects of the multiplication of algebraic expressions. The various terms used in the concept of the multiplication of various algebraic expressions, different types of algebraic equations, and their multiplication with the same kind of mathematical equation or the other categories of algebraic expressions, like the multiplication of monomial with monomial, binomial with binomial, polynomials with the monomials, or various other types of multiplication, are explained here.