Multiplication of Algebraic Expressions

Multiplication of algebraic expressions can be performed on algebraic expressions in the same way as it can be performed on two whole numbers or fractions. Multiplication of two algebraic expressions or variable expressions involves multiplying two expressions that are combined with arithmetic operations such as addition, subtraction, multiplication, division, and contain constants, variables, terms, and coefficients. In this article, let us learn about the rules of multiplication of algebraic expressions with solved examples.

Multiplication of 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 expressions, let's see some of the algebraic rules for multiplication.

(i) The product of two terms with the same signs is positive, and the product of two terms with different signs is negative.

For example:

• -a × - b = ab
• a × - b = -ab

(ii) We can add the exponents of the same bases in case of multiplication as a^m × aⁿ = a^m+n

For example:

• a² × a³ = a²⁺³ = a⁵
• b⁵ × b⁸ = b⁵⁺⁸ = b¹³

We use different algebraic identities for the multiplication of algebraic expressions to make the complex problem simple. The below-given image shows some of the helpful formulas and expansions to remember while performing multiplication of algebraic expressions:

The result of the multiplication of algebraic expressions is obtained by multiplying each term of the polynomial by the other and then taking the algebraic sum of these products. There are different types of multiplication of algebraic expressions:

• Monomial by monomial/binomial
• Binomial by Binomial
• Polynomial by monomial/binomial

In any case, we multiply each term of the first polynomial with each term of the second. Let us discuss them case by case.

Multiplication of Two Monomials

An algebraic expression is considered a monomial when it only contains one term like 5ab. Monomials usually include variables, numbers, or multiple numbers and/or variables that are multiplied together.

Product of two monomials = numerical coefficients × variable parts

Example: Find the product of 6ab and -3a²b³

Solution

6ab × -3a²b³

= 6 × -3 × ab × a²b³

= -18 × a¹⁺² × b¹⁺³

= -18a³b⁴

Multiplication of a Polynomial by a Monomial

An algebraic expression is considered a polynomial when it contains variables, coefficients, that involve only the operations of subtraction, addition, multiplication, and non-negative integer exponentiation of variables.

Multiply each term of the polynomial by the monomial, using the distributive law: a × (b + c) = a × b + a × c

Example: Find following product: 5a²b² × (3a² - 4ab + 6b²)

Solution

5a²b² × (3a² - 4ab + 6b²)

= (5a²b²) × (3a²) + (5a²b²) × (- 4ab) + (5a²b²) × (6b²)

= 15a⁴b² - 20a³b³ + 30a²b⁴

Multiplication of Two Binomials

An algebraic expression is considered binomial when it is made of the sum or difference of two terms. We multiply two binomials by using the distributive law of multiplication twice.

Let us find the product of two binomials (a + b) and (c + d).

(a + b) × (c + d)

= a × (c + d) + b × (c + d)

= a × c + a × d + b × c + b × d

= ac + ad + bc + bd

Note: This method is known as the horizontal multiplication method.

Example: Multiply (3a + 5b) and (5a - 7b).

Solution

(I) Horizontal multiplication method

(3a + 5b) × (5a - 7b)

= 3a × (5a - 7b) + 5b × (5a - 7b)

= (3a × 5a - 3a × 7b) + (5b × 5a - 5b × 7b)

= (15a² - 21ab) + (25ab - 35b²)

= 15a² - 21ab + 25ab - 35b²

= 15a² + 4ab - 35b²

(II) Column wise multiplication

\(3x + 5y\)

\(\times (5x - 7y)\)
_____________
\(15x^2 + 25xy\) ⇐ multiplication by 5x.
\( - 21xy - 35y^2\) ⇐ multiplication by -7y.
__________________
\(15x^2 + 4xy - 35y^2\) ⇐ Added the above terms.
__________________

IV. Multiplication by Polynomial

Example: Multiply (5x² – 6x + 9) with (2x -3)

Solution

\( 5x^2 – 6x + 9 \)

\(\times (2x - 3)\)
____________________
\(10x^3 - 12x^2 + 18x \) ⇐ multiplication by 2x.

\( - 15x^2 + 18x - 27\) ⇐ multiplication by -3
______________________
\(10x^3 – 27x^2 + 36x - 27 \) ⇐ Added the above terms.
______________________

Therefore (5x² – 6x + 9) × (2x - 3) is (10x³ – 27x² + 36x – 27)

Related Articles on Multiplication of Algebraic Expressions

Check out the following pages related to the multiplication of algebraic expressions

• Multiplying Exponents
• Rationalization
• Exponential Equations
• Two-Variable Linear Equations

Important Notes on Multiplication of Algebraic Expressions

Here is a list of a few points that should be remembered while studying the multiplication of algebraic expressions:

• We can add or subtract terms only if they are like terms.
• For division or multiplication of algebraic expressions, the terms can be either like terms or unlike terms.
• Multiplication of algebraic expressions can be done by using algebraic identities.
• Remember to distribute the negative sign also to all terms in the bracket.

What is Multiplication of Algebraic Expressions?

Multiplication of algebraic expressions can be performed on algebraic expressions in the same way as it can be performed on two whole numbers or fractions.

What Are the Types of Algebraic Expressions used in Multiplication of Algebraic Expressions?

There are mainly three types of algebraic expressions which include:

• Monomial Expression
• Binomial Expression
• Polynomial Expression

What are the Rules for Multiplication of Algebraic Expressions?

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.

How Do You Simplify Algebraic Expression in Multiplication of Algebraic Expressions?

The basic steps to simplify an algebraic expression in the multiplication of algebraic expressions are:

• Remove parentheses by using the distributive property.
• Use the exponent rules to multiply the terms with the same bases.
• Combine the like terms.

How To Do Multiplication of Algebraic Expressions?

Multiplication of algebraic expressions is done by using algebraic identities. Few algebraic identities are given below:

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (a + b)(a − b) = a² − b²
• (x+a)(x+b) = x² + x(a+b) +ab

How Do You Multiply Unlike Terms in Multiplication of Algebraic Expressions?

To multiply unlike terms in the multiplication of algebraic expressions :

• We multiply the numerical coefficients.
• We then multiply the variables.

How To Do Multiplication of Three Algebraic Expressions?

To multiply three algebraic expressions:

• We first multiply any two algebraic expressions.
• We then multiply this product by the third algebraic expression.