Multiplying Binomials

The algebraic expressions are classified into different types based on the numbers of terms, such as monomials, binomials, trinomials, and polynomials. Polynomials with exactly two terms, and combined by addition or subtraction, are called binomials. 2x + 3 and x2 + 2 are some examples of binomials.

Multiplying binomials is similar to multiplying numbers, but it involves variables instead of numbers. The key difference is that binomials involve multiplying algebraic expressions, requiring attention to variables and like terms. 

To multiply binomials, first multiply each term in the first binomial by each term in the second binomial and then combine the like terms using addition or subtraction. The methods of multiplying binomials are:

• Distributive Property

• FOIL Method

• Vertical Method

The distributive property involves multiplying each term inside the parentheses by the terms outside, and then combining the results. We can see how to multiply a binomial using the distributive property by the following steps:

Multiply (x + 2)(x + 3)

In distributive property, we need to multiply the first term of the first binomial with all the terms in the second binomial, and then the second term from the first binomial with all the terms in the second binomial.

Multiply x by (x + 3) = x² + 3x

Multiply 2 by (x + 3) = 2x + 6

Add both, the result: x² + 3x + 2x + 6 

Add the like terms: x² + 5x + 6

FOIL method is used to multiply the binomials. The word FOIL stands for First, Outer, Inner, and Last, it refers to the order in which the terms should be multiplied. 

Example: Multiply: (x + 2)(x + 3)

First: Multiply the first terms of both binomials. 

x × x = x²

Outer: Then multiply the first term of the first binomial by the second term of the second binomial.

x × 3 = 3x

Inner: Multiply the inner terms from the binomials, i.e., second term from the first binomial with first term in the second binomial. 

2 × x = 2x

Last: Multiply the last terms of binomials.

2 × 3 = 6

Combine the terms by adding the like terms.

x² + 3x + 2x + 6 = x² + 5x + 6

To multiply the binomials using the vertical method, we can use the following steps with the example, (x + 2)(x + 3).

Arrange the binomials one below the other.

Multiply each term:
x × x = x2
x × 3 = 3x
2 × x = 2x
2 × 3 = 6

Line up terms and add the like terms.

So the final answer is: x2 + 5x + 6 

Multiplying binomials is not only used in math, it helps to solve many problems in real-life situations. When we multiply binomials, we are finding areas, calculating profits, or solving physics problems. Listed below are some real-life applications of multiplying polynomials.

• Construction: Multiplying binomials is used to calculate the area of rectangular space. For example, if the length of the floor is (x + 2) meters and the width is (x + 3) meters, the area of the room will be their product, it can be used to estimate the materials required. 

• Business and Finance: In business and finance, multiplying binomials helps in pricing, budgeting, and maximizing profit.

• Data Analysis and Statistics: In statistics, binomials are used to create models and to make predictions. For example, a business might predict sales by using the binomials to estimate growth and change. 

• Game Designing: Developers use polynomial expressions like binomials to model movement of characters, calculate distances, or create animations. Multiplying expressions helps to simulate how fast something moves.