(a+b)(c+d)=ac+ad+bc+bd 証明

3 min read Jun 16, 2024

Proof of (a+b)(c+d)=ac+ad+bc+bd

This equation is a fundamental concept in algebra, often referred to as the distributive property. It states that the product of two binomials can be expanded as the sum of four terms.

Here's a breakdown of the proof:

1. Understanding the terms:

(a + b) and (c + d) are two binomials, each containing two terms.
ac, ad, bc, and bd are the four terms we aim to obtain after the expansion.

2. Applying the distributive property:

We start by distributing the first term of the first binomial (a) to both terms of the second binomial:

a(c + d) = ac + ad

Next, we distribute the second term of the first binomial (b) to both terms of the second binomial:

b(c + d) = bc + bd

3. Combining the results:

Finally, we add the two results from step 2 to get the complete expansion:

(a + b)(c + d) = ac + ad + bc + bd

Example:

Let's illustrate this with an example:

Suppose we have:

a = 2
b = 3
c = 4
d = 5

Using the equation, we can find the product:

(2 + 3)(4 + 5) = 2 * 4 + 2 * 5 + 3 * 4 + 3 * 5

Simplifying the equation:

5 * 9 = 8 + 10 + 12 + 15

45 = 45

Therefore, we have verified that the equation (a + b)(c + d) = ac + ad + bc + bd holds true.

Conclusion:

This proof demonstrates the distributive property in action, allowing us to expand the product of two binomials into a sum of four terms. This principle is fundamental in algebra and forms the basis for many other algebraic manipulations and calculations.