Multiplication of Algebraic Expressions(With Methods & Examples)

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In mathematics, addition, subtraction, multiplication, and division are four basic operations. Just like we multiply numbers we can perform the multiplication of algebraic expressions.  In the case of addition and subtraction, we can add or subtract only the like terms. But in the case of multiplication of algebraic expressions, participating terms don’t need to be the like terms. We can multiply two or more like terms as well as unlike terms.

Let’s understand the methods of multiplying algebraic expressions with steps and examples.

What is the Multiplication of Algebraic Expressions?

The multiplication of algebraic expressions is a method of multiplying two given expressions consisting of variables and constants. The general procedure involved in the multiplication of algebraic expressions is to 

• multiply the coefficients of the terms
• add the powers of the variables with the same base
• obtain the algebraic sum of the like and unlike terms

For example, $2x^{3} \times 4x^{2} = \left(2 \times 4 \right) \left(x^{3} \times x^{2} \right) = 8x^{3 + 2} = 8x^{5}$

$3a^{2} \times 5b^{3} = \left(3 \times 5 \right) \left(a^{2} \times b^{3} \right) = 15a^{2}b^{3}$

While multiplying the coefficients, the general rules of multiplication of integers are followed.

• $\text{Positive} \times \text{Positive} = \text{Positive}$, i.e., $+ \times + = +$
• $\text{Positive} \times \text{Negative} = \text{Negative}$, i.e., $+ \times – = -$
• $\text{Negative} \times \text{Positive} = \text{Negative}$, i.e., $- \times + = -$
• $\text{Negative} \times \text{Negative} = \text{Positive}$, i.e., $- \times – = +$

And, while multiplying variables, the multiplication rule of exponents is followed

• $a^{m} \times a^{n} = a^{m + n}$

How to Multiply Algebraic Expressions?

There are many types of algebraic expressions such as monomials, binomials, trinomials, polynomials, and multinomials. Depending on the type of algebraic expressions these are the different types of multiplication of algebraic expressions.

• Multiplying a monomial by a monomial
• Multiplying a monomial by a binomial
• Multiplying a binomial by a binomial
• Multiplying a monomial by a polynomial
• Multiplying a polynomial by a polynomial

Multiplying a Monomial by a Monomial

An algebraic expression is considered a monomial when it contains only one term, such as $8x^{3}$, $-2b^{4}$, etc. While multiplying a monomial by another monomial, the coefficient of the two monomials is multiplied separately and the variables are multiplied separately.

Let’s consider a few examples to understand the procedure of multiplying a monomial by a monomial.

Examples

Ex 1: $9y^{5} \times 4y^{3}$

$9y^{5} \times 4y^{3} = \left(9 \times 4 \right) \left(y^{5} \times y^{3} \right) $

$= 36y^{5 + 3} = 36y^{8}$

Ex 2: $-5a^{2} \times 3a^{5}$

$-5a^{2} \times 3a^{5} = \left(-5 \times 3)(a^{2} \times a^{5} \right)$

$= -15a^{2 + 5} = -15a^{7}$.

Ex 3: $12a^{2}b \times 7abc$

$12a^{2}b \times 7abc = \left(12 \times 7 \right) \left(a^{2} \times a \right) \left(b \times b \right) \left(c \right)$

$= \left(12 \times 7 \right) \left(a^{2} \times a^{1} \right) \left(b^{1} \times b^{1} \right) \left(c \right)$ 

$= 84 a^{2 + 1} b^{1 + 1}c = 84a^{3}b^{2}c$.

Multiplying a Monomial by a Binomial

An algebraic expression is considered a binomial when it contains two terms, such as $m^{2} – 3$, $2a^{2} + b^{3} $, $5x – 2y$, etc. While multiplying a monomial by a binomial, the monomial is multiplied with both the terms of the binomial separately, and the product is written by simplifying the terms.

Note: The two individual terms of a binomial taken separately are monomials and we use the process of multiplying two monomials while multiplying the terms of a binomial by a monomial.

Let’s consider a few examples to understand the procedure of multiplying a monomial by a binomial.

Examples

Ex 1: $6y^{2} \times \left(x^{3} + 2y^{2} \right)$

$6y^{2} \times \left(x^{3} + 2y^{2} \right) = 6y^{2} \times x^{3} + 6y^{2} \times 2y^{2} = 6x^{3}y^{2} + 12y^{4}$.

Ex 2: $\left(a^{3} – b^{3} \right) \times – 2a^{2}$

$\left(a^{3} – b^{3} \right) \times – 2a^{2} = a^{3} \times (- 2a^{2}) – b^{3} \times (- 2a^{2}) = -2a^{5} + 2a^{2}b^{3}$

Multiplying a Binomial by a Binomial

While multiplying a binomial by a binomial, each of the two terms of the first binomial is multiplied with each of the two terms of the other binomial. The final answer is obtained after simplifying each of the four monomial terms obtained.

This process of multiplying two binomials is called the FOIL method of multiplication.

Note: The FOIL Method is used to multiply binomials. 

FOIL is an acronym. The letters stand for First, Outer, Inner, and Last, referring to the order of multiplying terms. 

• $\text{F}$irst: Multiply the First term of the first binomial by the First term of the second binomial
• $\text{O}$uter: Multiply the Outer term of the first binomial by the Outer term of the second binomial
• $\text{I}$nner: Multiply the Inner term of the first binomial by the Inner term of the second binomial
• $\text{L}$ast: Multiply the Last term of the first binomial by the Last term of the second binomial

You multiply the first terms, then outside terms, then inside terms, last terms, and then combine like terms for your answer.

Let’s consider a few examples to understand the procedure of multiplying a binomial by a binomial.

Examples

Ex 1: $\left(2x + 3y \right) \times \left(3x – 2y \right)$

$= 2x \times 3x + 2x \times \left(- 2y \right) + 3y \times 3x + 3y \times \left(-2y \right)$

$= 6x^{2} – 4xy + 9xy – 6y^{2} = 6x^{2} + 5xy – 6y^{2}$

Ex 2: $\left(l^{2} – 2l \right) \times \left(2l^{2} + 4 \right)$

$= l^{2} \times 2l^{2} + l^{2} \times 4 – 2l \times 2l^{2} – 2l \times 4$

$= 2l^{4} + 4l^{2} – 4l^{3} – 8l = 2l^{4} – 4l^{3} + 4l^{2} – 8l$

Multiplying a Monomial by a Polynomial

An algebraic expression is considered a polynomial when it contains one or more terms, such as $2x^{4} – 3x^{3} + 5x^{2} + 7x + 9$, $3x^{2} + 2y^{2} + z^{2} + 2xy – 5yz + 6zx$, $a^{3} – 3a^{2}b + 3ab^{2} – b^{3}$, etc. While multiplying a monomial by a polynomial, the monomial is multiplied with every term of the polynomial separately and the product is written by simplifying the terms.

Let’s consider a few examples to understand the procedure of multiplying a monomial by a polynomial.

Examples

Ex 1: $2x^{2} \times \left(5x^{3} – 2x^{2} + 3x + 9 \right)$

$= 2x^{2} \times \left(5x^{3} – 2x^{2} + 3x + 9 \right) = 2x^{2} \times 5x^{3} + 2x^{2} \times (- 2x^{2}) + 2x^{2} \times 3x + 2x^{2} \times 9$

$= 10x^{5} – 4x^{4} + 6x^{3} + 18x^{2}$

Ex 2: $(5x^3 – 12x + 9) \times 7x^{3} = 5x^3 \times 7x^{3} – 12x \times 7x^{3} + 9 \times 7x^{3} = 35x^6 – 84x^{4} + 63x^{3}$

Multiplying a Polynomial by a Polynomial

While multiplying a polynomial by a polynomial, every term of the first polynomial is multiplied by every term of the other polynomial the product is written by simplifying the terms.

Let’s consider a few examples to understand the procedure of multiplying a polynomial by a polynomial.

Examples

Ex 1: $\left(2x^{2} + 3x + 5 \right) \times \left(3x^{3} – 4x^{2} + 5x – 9 \right)$

$ = 2x^{2} \times \left(3x^{3} – 4x^{2} + 5x – 9 \right) + 3x \times \left(3x^{3} – 4x^{2} + 5x – 9 \right) + 5 \times \left(3x^{3} – 4x^{2} + 5x – 9 \right)$

$ = 2x^{2} \times 3x^{3} + 2x^{2} \times \left(- 4x^{2} \right) + 2x^{2} \times 5x + 2x^{2} \times \left(- 9 \right) + 3x \times 3x^{3} +$

$ 3x \times \left(- 4x^{2} \right) + 3x \times 5x + 3x \times \left(- 9 \right)  +  5 \times 3x^{3}+ 5 \times  \left(-4x^{2} \right) + 5 \times  5x + 5 \times  \left(-9 \right)$

$ = 6x^{5} – 8x^{4} + 10x^{3} – 18x^{2} + 9x^{4} – 12x^{3} + 15x^{2} – 27x + 15x^{3} – 20x^{2} + 25x – 45$

$ = 6x^{5} – 8x^{4} + 9x^{4} + 10x^{3} – 12x^{3} + 15x^{3} – 18x^{2} + 15x^{2} – 20x^{2} – 27x + 25x – 45$

$ = 6x^{5} + x^{4} + 13x^{3} – 23x^{2} – 2x – 45$

Tips for Multiplication of Algebraic Expressions

• We can multiply any algebraic term with any other algebraic term. It can be the product of two like terms or a product of like and unlike terms.
• We can ignore the order of variables in like terms in an algebraic expression. For example,  $3a + 2b$, and, $9b + a$ both are like terms.
• We can ignore writing $1$ as the numerical coefficient of any term. For example, $xy$ is the same as $1xy$.
• We can replace a missing term with $0$ with the same variables. For example, a missing term can be written as $0x$, $0y$, or $0xy$ depending on the variables of the missing term.

Practice Problems

Multiply the following

• $2x^{3} \times 6x^{2}$
• $-3x^{2} \times 5y^{2}$
• $2x^{3} \times \left(3a^{2} + 2b^{2} \right)$
• $\left(5l + 6m \right) \times \left(7m – 2l \right)$
• $\left(3y^{3} – 27 \right) \times \left(2y^{2} + 4 \right)$
• $8a^{2} \times \left(5x^{2} – 2xy + 3y^{2} \right)$
• $\left(a – 2b + 3c \right) \times \left(3x – 2y + z \right)$
• $\left(3x^{3} – 2x^{2} + x – 7  \right) \times \left(9x^{3} + 4x^{2} + 3 \right)$

FAQs

Conclusion

The multiplication of algebraic expressions is a method of multiplying two given expressions consisting of variables and constants. The general procedure involved in the multiplication of algebraic expressions is to 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.

Recommended Reading

• Division of Algebraic Expressions(With Methods & Examples)
• Subtraction of Algebraic Expressions(With Methods & Examples)
• Addition of Algebraic Expressions(With Methods & Examples)
• What is Algebraic Expression(Definition, Formulas & Examples)
• What is Algebra – Definition, Basics & Examples
• What is Pattern in Math (Definition, Types & Examples)
• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties
• Rationalize The Denominator(With Examples)
• Multiplication of Irrational Numbers(With Examples)