Multiplication can be performed on algebraic expressions the same way as it can be performed on two whole numbers or fractions. Multiplication of algebraic expressions or variable expressions involves multiplying two expressions combined with arithmetic operations such as addition, subtraction, multiplication, division and contain constants, variables, terms, and coefficients.

The multiplication of the algebraic expressions are divided into 3 types namely, monomial by a monomial or binomial or trinomial, binomial by binomial, polynomial by monomial or binomial. In this article, we will discuss about the multiplication of algebraic expressions in detail. Scroll down to learn more!

Multiplication of Algebraic Expression

Multiplying algebraic expressions involves the following steps. Multiply the coefficients of the terms, add the powers of the variables with the same base, and obtain the algebraic sum of the like and unlike terms before learning about the multiplication of algebraic expression.

1. The product of two terms with the same signs is positive, and two terms with different signs is negative.
   Example:
   \(-a \times -b = ab\)
   \(a \times -b = -ab\)
2. We can add the exponents of the same bases in case of multiplication as \(a^m \times a^n = a^{m+n}\)
   Example:
   \(a^2 \times a^3 = a^{2+3} = a^5\)
   \(b^5 \times b^8 = b^{5+8} = b^{13}\)

What are Algebraic Expressions?

Definition: An algebraic expression is the combination of the constants and the variables connected by the fundamental operators like addition \((+)\), subtraction \((-)\), etc.
Examples: \(4x + 5,\, 10y – 5\) are examples of algebraic expressions.

Terms Used in Algebra

The terms used in algebra are given below:

1. Variables: The symbol that makes various numerical values is known as a variable.

Examples:
(i) The perimeter \(P\) of a square of side \(S\) is given by the formula, \(P = 4 \times S.\) Now, the number \(4\) is the constant, while \(P\) and \(S\) are the variables.
(ii) The circumference \(C\) of the circle of the radius \(R\) is given by a formula, \(C = 2 \pi R.\) Here, \(2\) and \(\pi\) are constants while \(C\) and \(R\) are variables.

2. Coefficients or constants: A symbol having a fixed numerical value is called a constant, and the term of the expression having no literal factor is known as the constant term.
Examples: In the given algebraic expression \(x^2 + 5x – 3\), the constant term is \(-3.\)

3. Monomial: Monomial is an expression with one term.
Example: \(4x,\,5y,\,7z\)

4. Binomial: Binomial is an expression with two terms.
Example: \(4x + 4,\,5y + y,\,7z + z\)

5. Trinomial: Trinomial is an expression with three terms.
Example: \(4x^2 + 4x +1\)

Polynomial Expression

Let \(x\) be a variable, \(n\) be a positive integer and as \(a_1,\,a_2,\,……,a_n\) be constants (real numbers).
Then \(f(x) = a_n x^{n} +a_{n-1} x^{n-1} + …… +a_1 x + a0\) is called the polynomial in variable \(x\).

Definition: A polynomial is the combination of the constants and the variables connected by the operations addition \((+)\), subtraction \((−)\), etc.
Examples: \(7x + 3,\,11y – 6\) are examples of Polynomials.

Multiplying Algebraic Expressions

In the multiplication of the algebraic expressions, you will use the following rules of sign:

1. The product of the two factors with the like signs is positive, and the product of two factors with unlike signs is negative.
   i.e., \( (+) \times (+) = + \)
   \( (+) \times (-) =\ – \)
   \( (-) \times (+) =\ – \)
   and \( (-) \times (-) = + \)
2. If \(a\) is any variable and \(m,\,n\) are the positive integers, then \(a^m \times a^n = a^{m+n}\)
   Example: \({a^3} \times {a^5} = {a^{3 + 5}} = {a^8},\;{y^4} \times y = {y^{4 + 1}} = {y^5}\) etc.

When two algebraic expressions are multiplied, the result is called the product, and the two expressions making up the product are called factors and multiplicands. The simplest case of multiplication is the multiplication of one monomial with another monomial. So, we first discuss the multiplication of two monomials.

How Do You Multiply Algebraic Expressions?

The product of the multiplication of the algebraic expressions is acquired by multiplying each term of the polynomial by the other and then obtaining the algebraic sum of these products. There are various types of multiplication of the algebraic expressions, which are given below:

1. Monomial by monomial or binomial or trinomial.
2. Binomial by Binomial.
3. Polynomial by monomial or binomial.

In any case, you can multiply each term of the first polynomial with each term of the second polynomial.  

Multiplication of Monomial by Monomial

Consider the multiplication of two monomials, say \(3ab\) and \(5b\). We shall multiply these two monomials by the repeated use of commutativity and associativity of multiplication.
We have two rules for the multiplication of two monomials:
Rule 1: A coefficient of a product of two monomials is equal to the product of their coefficients.
Rule 2: The variable part in a product of two monomials is equal to the product of the variable parts in the given monomials.
These two rules are also applicable for the product of three or more monomials.

Multiplication of Monomial and a Binomial

In operations of rational numbers, you have learnt the distributivity of multiplication over addition. This property helps us to simplify the expressions of form \(3 \times (2 + 7).\)
Multiplication of literals is also distributive over their addition, i.e., \(a \times (b + c) = a \times b + a \times c.\) This property is used to multiply a monomial and a binomial.
Thus, if \(P,\,Q\) and \(R\) are three monomials, then we have

1. \( P \times (Q + R) = (P \times Q) + (P \times R)\)
2. \((Q + R) \times P = Q \times P + R \times P\)

In other words, we multiply each term of the binomial \(Q + R\) by the monomial \(P\) and add the products to get the required result.
Also, \( P \times (Q – R) = (P \times Q) – (P \times R)\) and \((Q – R) \times P = (Q \times P) – (R \times P)\)

Multiplication of a Monomial by a Trinomial

When we multiply the monomial and the trinomial, we get the product as the trinomial.
Example: When we multiply \(5a\) and \((a + b +c)\)
Here, \(5a\) is the monomial, and \((a + b +c)\) is the trinomial.
When we write the multiplication expression, we get \(5a(a + b + c).\)
Use the distributive law and then multiply monomial with every term of the trinomial, and then this will be done in the given way:
\(= (5a \times a) + (5a \times b) + (5a \times c)\)
\(= 5a^2 + 5ab + 5ac\)
Hence, the product we got is \(5a (a + b + c) = 5a^2 + 5ab + 5ac.\)

Multiplication of Two Binomials

In the multiplication of two binomials, we will use the distributive property of multiplication of literals over their additions, as discussed below.
Consider two binomials, say \((a + b)\) and \((c + d)\). By using the distributive property of multiplication of literals over their addition, we have,
\((a + b) \times (c + d) = a \times (c + d) + b \times (c + d)\)
\(= (a \times c + a \times d) + (b \times c + b \times d)\)
\(= ac + ad + bc + bd\)
It follows from the above result that to multiply any two binomials, we multiply each term of one binomial by each term of the other and add the products.

Multiplying Monomials and Polynomials

An algebraic expression is considered a polynomial when it contains variables, coefficients that involve only the operations of subtraction, addition etc.
You can multiply each polynomial term by the monomial by using the distributive law:
\(a \times (b + c) = a \times b + a \times c.\)
Example: Find the product: \(5a^2 b^2 \times (3a^2 – 4ab + 6b^2)\)
\(5a^2 b^2 \times (3a^2 – 4ab + 6b^2)\)
\( = (5a^2 b^2) \times (3a^2) + (5a^2 b^2) \times (- 4ab) + (5a^2 b^2) \times (6b^2)\)
\( = 15a^4 b^2 – 20a^3 b^3 + 30a^2 b^4\)

Solved Examples on Multiplication of Algebraic Expressions

Q 1. Find the value of \(\left( {5{a^6}} \right) \times \left( { – 10a{b^2}} \right) \times \left( { – 2.1{a^2}{b^3}} \right)\) for \(a = 1\) and \(b = \frac{1}{2}.\)
Ans: We have,
\(\left( {5{a^6}} \right) \times \left( { – 10a{b^2}} \right) \times \left( { – 2.1{a^2}{b^3}} \right)\)
\( = \left( {5 \times – 10 \times – 2.1} \right) \times \left( {{a^6} \times a \times {a^2} \times {b^2} \times {b^3}} \right)\)
\( = \left( {5 \times – 10 \times – \frac{{21}}{{10}}} \right) \times \left( {{a^6} \times a \times {a^2} \times {b^2} \times {b^3}} \right)\)
\( = 105a^{6 + 1 + 2} b^{2 + 3} = 105a^9 b^5\)
Putting \(a = 1\) and \(b = \frac{1}{2}\), we have
\(105{a^9}{b^3} = 105 \times {(1)^9} \times {\left( \frac{1}{2} \right)^5} = 105 \times 1 \times \frac{1}{{32}} = \frac{105}{32}.\)

Q.2. Multiply: \(2x\) by \(3x + 5y\)
Ans: We have,
\(2x \times \left( {3x + 5y} \right) = 2x \times 3x + 2x \times 5y = 6{x^2} + 10xy.\)

Q.3. Multiply: \(- 8a{b^2}c,\;3{a^2}b\) and \(\frac – {1}{6}.\)
Ans: We have:
\(\left( { – 8{\text{a}}{{\text{b}}^2}{\text{c}}} \right) \times \left( {3{{\text{a}}^2}{\text{b}}} \right) \times \left( { – \frac{1}{6}} \right)\)
\( = \left( { – 8 \times 3 \times \frac{{ – 1}}{6}} \right) \times \left( {{\text{a}} \times {{\text{a}}^2} \times {{\text{b}}^2} \times {\text{b}} \times {\text{c}}} \right)\)
\( = 4{{\text{a}}^{\left( {1 + 2} \right)}} \times {{\text{b}}^{\left( {2 + 1} \right)}} \times {\text{c}} = 4{{\text{a}}^3}{{\text{b}}^3}{\text{c}}.\)

Q.4. Multiply: \((3x + 2y)\) and \((5x + 3y).\)
Ans: We have,
\((3x + 2y) + (5x + 3y)\)
\( = 3x \times \left( {5x + 3y} \right) + 2y \times \left( {5x + 3y} \right)\)
\( = \left( {3x \times 5x + 3x \times 3y} \right) + \left( {2y \times 5x + 2y \times 3y} \right)\)
\( = \left( {15{x^2} + 9xy} \right) + \left( {10xy + 6{y^2}} \right)\)
\( = 15{x^2} + 9xy + 10xy + 6{y^2}\)
\( = 15{x^2} + 19xy + 6{y^2}.\)

Q.5. Find the value of the given product: \(\left( {x + 2y} \right)\left( {x – 2y} \right)\) at \(x = 1,\,y = 0.\)
Ans: We have,
\(\left( {x + 2y} \right)\left( {x – 2y} \right)\)
\( = x\left( {x – 2y} \right) + 2y\left( {x – 2y} \right)\)
\( = x \times x – x \times 2y + 2y \times x – 2y \times 2y\)
\( = {x^2} – 2xy + 2yx – 4{y^2}\)
\( = {x^2} – 4{y^2}\)
When \(x = 1,\,y = 0\) we get
\(\left( {x + 2y} \right)\left( {x – 2y} \right)\)
\( = {x^2} – 4{y^2} = {(1)^2} – 4 \times {(0)^2} = 1 – 0 = 1.\)

Summary

The multiplication of the algebraic expressions is defined as the method of multiplying two given expressions which involves the variables and the constants. When two algebraic expressions are multiplied, the product is called the product, and the two expressions making up the product are called factors and multiplicands. The multiplication of the algebraic expressions are divided into various types namely, multiplication of monomial by a monomial, multiplication of monomial and a binomial, multiplication of a monomial by a trinomial, multiplication of two binomials, and multiplying monomials and polynomials.

FAQs

Q.1. What is the rule of multiplication in algebra?
Ans: In the multiplication of the algebraic expressions, we will use the following rules of sign:
1. The product of the two factors with the like signs are positive, and the product of two factors with unlike signs is negative.
i.e., \( (+) \times (+) = + \)
\( (+) \times (-) =\ – \)
\( (-) \times (+) =\ – \)
and \( (-) \times (-) = + \)
2. If \(a\) is any of the variable and \(m,\,n\) are the positive integers, then \(a^m \times a^n = a^{m+n}\)
Example: \({a^3} \times {a^5} = {a^{3 + 5}} = {a^8},\,\;{y^4} \times y = {y^{4 + 1}} = {y^5}\) etc.

Q.2. What is an example of a multiplication expression?
Ans: The multiplication of the algebraic expressions is the method of multiplying two given expressions consisting of the variables and the constants.
Example: \(4xy + 9\) in this expression \(x\) and \(y\) are the variables, whereas \(4\) and \(9\) are the constants.

Q.3. How do you write an expression for multiplication?
Ans: The expression \(4x + 5\) is acquired from the variable \(x\) first by multiplying \(x\) by the constant \(4\) and then adding the constant \(5\) to the product. Similarly, \(10y – 20\) is acquired by first multiplying \(y\) by \(10\) and then subtracting \(20\) from the product.
The expression \(x^2\) is acquired by multiplying the variable \(x\) by itself.
\(x \times x = x^2\)
Just as \(4 \times 4\) is written as \(4^2\), we write \(x \times x = x^2\). It is commonly read as \(x\) squared.

Q.4. What are the three rules of algebra?
Ans: The three rules of algebra are given below:
1. Commutative rule
2. Associative rule
3. Distributive rule

Q.5. What are the two basic rules for solving algebraic equations?
Ans: The two basic rules for solving the equations are the addition rule and the multiplication or the division rule.

Q.6. How do you simplify a multiplication problem?
Ans: The multiplication expression is the mathematical expression with multiplication. So, to simplify the expression, you have to combine the numbers together and then the like variables together. For the variables, you write each variable to a certain power. This power tells you how many of those variables you are multiplying together.

We hope this detailed article on the multiplication of algebraic expressions helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you.