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 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.
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.
Binomial expression: The polynomial expression involves two terms, for example, 2x - 1, xy - 5z2, etc.
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:
a*a = a2
2a*2b = (2*2)*(a*b) = 4ab
6ab*3x = (6*3)*(ab*x) = 18abx
5xy * 4x2 * 2x3 = (5*4*2)*{x(1+2+3) * y} = 40yx6
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:
a*(a +b) = a2 +ab
4xy(3xy) = 12(xy)²
(-ab)*(a –b +c) = -a2b +ab2 –abc
3xyz(x +2y -3) = 3yzx2 +6xzy2 -3xyz
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.
FAQs on Multiplication of Algebraic Expression
1. What are the practical uses of the 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 expressions (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 that the operations used in the algebraic expressions for learning go through the given multiplication of algebraic expression examples.
2. What are some of the common mistakes made while doing the multiplication of algebraic expressions?
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.
3. What are algebraic expressions and what types of algebraic expressions are used in the multiplication of algebraic expressions?
Algebraic expressions are mathematical expressions containing the combination of the mathematical constant and the variables connected by one or many mathematical operations from the four fundamental mathematical operators, the fundamental mathematical expressions are addition (+), subtraction (-), multiplication (), and division (). Some examples of Algebraic Expressions are - 4 x + 8, x - y, etc.
In the example 4 x + 8, the numbers 4 and 8 are the provided constant of the expression, and the letter x will be the variable of the equation.
In the concept of multiplication, the Algebraic expressions are mainly divided into three different types -
Monomial Algebraic Expression, an example of it is - 6 a.
Binomial Algebraic Expression, an example of it is - x + 2 y
Polynomial Algebraic Expression, an example of it is - 3x2 - 4xy + 6y2.
4. How can the student simplify a multiplication problem or expressions?
The basic process or steps to simplifying an algebraic expression in the topic of multiplication of algebraic expressions are as follows-
Remove parentheses present with the help of the distributive property of algebraic expressions.
To multiply two or more variables with the same base, use the exponent rules to get the right power for the result.
Identify and combine all the like terms, if any.
Use the various algebraic identities, where they are applicable.
There are many questions that are asked in the examination on the simplification of a mathematical problem or expressions. The student can prepare them with a lot of practice, the Sample questions available at the official website of Vedantu are a great way to practice any subject of maths.
5. What are the important points to remember on the topic of Multiplication of Algebraic Expression?
Listed down below are all of the important notes or points to remember while studying the concept of the multiplication of various algebraic expressions.
One can only perform the operation of addition or the subtraction between two like terms, so the student has to identify the like terms and solve those like terms first.
But for the mathematical operation of multiplication and division, the term can either be like or unlike terms. The, unlike terms, do not have any issue in the multiplication or division.
If the variables have the same base, then for the multiplication and division, the student can use the exponent rules.
The students should give special attention to the negative sign, like when opening a bracket the negative sign will be distributed among all the terms inside the bracket.
Remember all the algebraic expressions as they can be used to easily simplify the various mathematical expressions.