(a-b)^5 Math Formula

Suzhoumeite

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Jul 03, 2024

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(a-b)^5: The Formula and Its Expansion

In algebra, the formula for raising a binomial to a power is a fundamental concept. One of the most common and useful formulas is the expansion of (a-b)^5. In this article, we will explore the formula, its expansion, and some examples to illustrate its application.

The Formula

The formula for (a-b)^5 is given by:

(a-b)^5 = a^5 - 5a^4b + 10a^3b^2 - 10a^2b^3 + 5ab^4 - b^5

This formula is an extension of the binomial theorem, which states that for any positive integer n, the expansion of (a+b)^n is given by:

(a+b)^n = a^n + na^(n-1)b + n(n-1)a^(n-2)b^2 + ... + b^n

By substituting b with -b in the binomial theorem, we get the formula for (a-b)^n. For n=5, we get the formula above.

Expansion

To understand the formula, let's break it down:

• a^5 is the first term, which is the fifth power of a.
• -5a^4b is the second term, which is the product of a to the power of 4 and -b.
• 10a^3b^2 is the third term, which is the product of a to the power of 3 and b squared.
• -10a^2b^3 is the fourth term, which is the product of a squared and -b cubed.
• 5ab^4 is the fifth term, which is the product of a and b to the power of 4.
• -b^5 is the sixth and final term, which is the fifth power of -b.

Examples

1. Simple Expansion

Let's expand (x-2)^5 using the formula:

(x-2)^5 = x^5 - 5x^4(2) + 10x^3(2^2) - 10x^2(2^3) + 5x(2^4) - 2^5

= x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32

2. Simplifying Expressions

Simplify the expression (a-b)^5 + (a+b)^5:

(a-b)^5 + (a+b)^5

= (a^5 - 5a^4b + 10a^3b^2 - 10a^2b^3 + 5ab^4 - b^5) + (a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5)

= 2a^5 + 20a^3b^2

In conclusion, the formula for (a-b)^5 is a powerful tool for expanding and simplifying algebraic expressions. By understanding the formula and its expansion, you can tackle complex problems with ease and confidence.