Multiplication of Algebraic Expression

Algebraic expressions are polynomial equations used in algebra that are used for variety of purposes. Multiplication of algebraic expression is useful for various purposes and it is achieved by multiplying each term of algebraic equation with other algebraic expression

In this article, we have discuss the concept of multiplication of algebraic expressions, exploring its importance, rules, methods, and practical applications.

What are Algebraic Expressions?

Algebraic expressions are mathematical expressions containing variables, constants, and arithmetic operations like addition, subtraction, multiplication, and division. They can represent quantities and relationships in various mathematical contexts.

Components of Algebraic Expressions

Algebraic expressions consist of terms, which are the building blocks comprising variables, constants and coefficients combined with arithmetic operations.

Components of Algebraic Expressions are,

• Variables: Symbols representing unknown or changing values, typically denoted by letters like x or y.
• Coefficients: Numerical factors multiplying variables.
• Constants: Fixed numerical values, such as 2 or -5.
• Operators: Symbols indicating mathematical operations like addition, subtraction, multiplication and division.

This is explained as, 2x² – 7x + 11 = 0

In equation given above,

• Variable = x
• Coefficient of x² = 2
• Coefficient of x = -7
• Constant = 11

What is Multiplication of Algebraic Expressions?

Multiplication of algebraic expressions involves combining terms and simplifying the resulting expression. It involves combining terms using the distributive property and multiplying like terms together.

Below listed are rules and methods involved for Multiplication of Algebraic Expressions:

Rules for Multiplying Algebraic Expressions

• Multiplication of Two Monomials: Multiply the coefficients and add the exponents of like variables. In other words, multiply the coefficients together and combine the variables by adding their exponents.
• Multiplication of Two Binomials: Use FOIL method (First, Outer, Inner, Last) to distribute and combine like terms. Use the distributive property to multiply each term of one binomial by each term of the other binomial and then combine like terms.
• Multiplication of Monomial and Binomial: Distribute the monomial to each term in the binomial.
• Multiplication of Two Polynomials: Apply the distributive property to multiply each term of one polynomial by each term of the other.

FOIL Method for Binomials

FOIL method simplifies the multiplication of two binomials by multiplying the First terms, Outer terms, Inner terms and Last terms and then combining like terms.

Distributive Property in Multiplication

The distributive property allows us to distribute a factor to each term inside parentheses, simplifying expressions.

Examples of Multiplying Algebraic Expressions

• (2x + 3)(x – 5)

= 2x(x – 5) + 3(x – 5)

= 2x² – 10x + 3x – 15

= 2x² -7x – 15

• (a + 2b)(a – 3b)

= a(a – 3b) + 2b(a – 3b)

= a² – 3ab + 2ab – 6b²

= a² -ab – 6b²

How To Do Multiplication of Algebraic Expressions?

To perform multiplication of algebraic expressions:

• Identify like terms.
• Apply the distributive property when multiplying binomials or a monomial by a polynomial.
• Combine like terms after multiplication.

Multiplication of Two Monomials

When multiplying two monomials, multiply the coefficients together and combine the variables by adding their exponents.

Example: Multiply 3x² and 4x³.

Solution:

(3x²)(4x³) = 12x^{2+3} = 12x⁵

Multiplication of a Polynomial by a Monomial

To multiply a polynomial by a monomial, distribute the monomial across each term of the polynomial.

Example: Multiply 2x and 3x² + 4x + 1.

Solution:

2x(3x² + 4x + 1) = 6x³ + 8x² + 2x

Multiplication of Two Binomials

When multiplying two binomials, apply the distributive property by multiplying each term of one binomial by each term of the other binomial and then combine like terms.

Example: Multiply x – 2 and 3x – 1.

Solution:

(x + 2)(3x – 1) = 3x² + 5x – 2

Multiplication of Two Polynomials

Multiplying by a polynomial involves distributing each term of the polynomial across the terms of the other expression then combining like terms. This is explained by the example added below as,

Example: Multiply 2x² + x – 11 and 3x² + 4x – 1.

Solution:

(2x² + x – 11)(3x² + 4x – 1)

= 2x²(3x² + 4x – 1) + x(3x² + 4x – 1) – 11(3x² + 4x – 11)

= 6x⁴ + 8x³ – 2x² + 3x³ + 4x² -x – (33x² + 44x – 121)

= 6x⁴ + 8x³ + 3x³ – 2x² + 4x² – 33x² – 44x + 121

= 6x⁴ + 11x³ – 31x² – 44x + 121

Conclusion on Multiplication of Algebraic Expression

Multiplication of algebraic expressions is a fundamental concept in algebra enabling us to solve equations, simplify expressions and help us to solve various mathematical problems.

Solved Examples on Multiplication of Algebraic Expression

Example 1: Multiply the given algebraic expression (3x + 2)(4x – 5).

Solution:

Using FOIL Method:

(3x + 2)(4x − 5)

⇒ 3x⋅4x + 3x⋅(−5) + 2⋅4x + 2⋅(−5)

⇒ 12x² − 15x + 8x − 10

⇒ 12x² − 7x − 10

Simplified algebraic expression = 12x² − 7x − 10

Example 2: Find the product of given algebraic expression (a – 2)(a + 3).

Solution:

Using FOIL Method: 

(a − 2)(a + 3)

⇒ a⋅a + a⋅3 − 2⋅a − 2⋅3

⇒ a² + 3a − 2a − 6

⇒ a² + a − 6

Simplified algebraic expression = a² + a − 6

Example 3: Multiply given algebraic expression: (2x² – 3)(x + 4).

Solution:

Using FOIL Method:  

(2x² − 3)( x + 4)

⇒ 2x²⋅x + 2x²⋅4 − 3⋅x − 3⋅4

⇒ 2x³ + 8x² − 3x − 12

Simplified algebraic expression = 2x³ + 8x² − 3x − 12

Example 4: Multiply: (5a – 2b)(3a + b).

Solution:

Using FOIL Method:

(5a − 2b)(3a + b)

⇒ 5a⋅3a + 5a⋅b−2b⋅3a−2b⋅b

⇒ 15a² + 5ab − 6ab −2b²

⇒ 15a² − ab − 2b²

Simplified algebraic expression = 15a² − ab − 2b²

Example 5: Find the product of (2x + 1)(2x – 1).

Solution:

Using FOIL Method:

(2x + 1)(2x − 1)

⇒ 2x⋅2x + 2x⋅(−1) + 1⋅2x −1⋅1

⇒ 4x² − 2x + 2x − 1

⇒ 4x² − 1

Simplified algebraic expression = 4x² − 1

Practice Questions on Multiplication of Algebraic Expression

Q1: Simplify 3x(2x + 4).

Q2: Find the product of (5a – 2)(a + 3).

Q3: Multiply: (3x + 2)(4x – 5).

Q4: Calculate: (2a – 3)(a – 1).

Q5: Find the product of (a – 7)(a + 6).

FAQs on Multiplying Algebraic Expressions

What are Algebraic Expressions?

Algebraic expressions are mathematical expressions containing variables, constants, and arithmetic operations.

Why is Multiplying Algebraic Expressions Important?

Multiplication of algebraic expressions is essential for solving equations, simplifying expressions, and solving real-world problems.

How to Simplify Algebraic Expressions before Multiplication?

Simplify by combining like terms and applying the distributive property.

Can Algebraic Expressions Contain Variables?

Yes, algebraic expressions often contain variables which represent unknown quantities or variables.

What are Key Components of Algebraic Expressions?

Key components include variables, constants, coefficients and arithmetic operations.

What is Difference Between Monomials and Binomials?

Monomials have one term while binomials have two terms.

How Does FOIL Method Work?

FOIL method simplifies the multiplication of two binomials by multiplying the First, Outer, Inner and Last terms.

What is Purpose of Multiplying Polynomials?

Multiplying polynomials allows us to find the area of rectangles, solve geometric problems and model real-world situations.