(a-b)^5

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Jul 03, 2024

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(a-b)^5: A Binomial Expansion

In algebra, the binomial theorem is a powerful tool for expanding powers of binomials, which are expressions consisting of two terms. One of the most common binomial expansions is (a-b)^5, which is the focus of this article.

What is Binomial Expansion?

Binomial expansion is a method of expanding an expression of the form (x+y)^n or (x-y)^n, where x and y are variables and n is a non-negative integer. The resulting expansion is a sum of terms, each involving a combination of x and y raised to various powers.

The Formula for (a-b)^5

Using the binomial theorem, we can expand (a-b)^5 as follows:

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

This expansion can be derived using the binomial theorem formula:

(x+y)^n = x^n + nx^(n-1)y + (n(n-1)/2)x^(n-2)y^2 + ... + y^n

In this case, we substitute x = a and y = -b, and then apply the formula to get the above expansion.

Understanding the Expansion

Let's break down the expansion of (a-b)^5:

• The first term, a^5, is the result of raising a to the power of 5.
• The second term, -5a^4b, arises from the combination of 5 as and 1 b, with a negative sign.
• The third term, 10a^3b^2, comes from the combination of 3 as and 2 bs.
• The fourth term, -10a^2b^3, involves 2 as and 3 bs, with a negative sign.
• The fifth term, 5ab^4, is the result of combining 1 a and 4 bs.
• The final term, -b^5, is simply b raised to the power of 5, with a negative sign.

Applications of (a-b)^5

The expansion of (a-b)^5 has numerous applications in mathematics, physics, engineering, and computer science. Some examples include:

• Algebraic manipulations: The expansion can be used to simplify complex algebraic expressions involving a and b.
• Geometry and trigonometry: (a-b)^5 appears in formulas for areas and volumes of various geometric shapes.
• Calculus: The expansion is useful in calculating derivatives and integrals of functions involving a and b.
• Computer science: The formula has applications in algorithms and data structures, such as in the analysis of recursive functions.

In conclusion, the expansion of (a-b)^5 is a fundamental concept in algebra and has far-reaching implications in various fields.