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