%5E3-expand.jpeg "(x-1)^3 Expand")
(x-1)^3 Expansion
In algebra, expanding an expression involving exponents and binomials can be a bit tricky. One common expansion is the cube of a binomial, specifically, (x-1)^3. In this article, we will learn how to expand this expression using the binomial theorem.
The Binomial Theorem
The binomial theorem is a mathematical formula for expanding powers of a binomial, which is an expression consisting of two terms. The theorem states that:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where n is a positive integer, a and b are variables, and k is an integer that ranges from 0 to n. The binomial coefficient binom(n,k) is calculated as:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Expanding (x-1)^3
Using the binomial theorem, we can expand (x-1)^3 as follows:
$(x-1)^3 = \sum_{k=0}^3 \binom{3}{k} x^{3-k} (-1)^k$
Let's calculate each term:
k=0:$\binom{3}{0} x^3 (-1)^0 = x^3$k=1:$\binom{3}{1} x^2 (-1)^1 = -3x^2$k=2:$\binom{3}{2} x^1 (-1)^2 = 3x$k=3:$\binom{3}{3} x^0 (-1)^3 = -1$
Now, combine the terms:
$(x-1)^3 = x^3 - 3x^2 + 3x - 1$
Conclusion
In this article, we have successfully expanded (x-1)^3 using the binomial theorem. The expansion is a cubic expression with four terms. This expansion is useful in various algebraic manipulations and applications in mathematics, physics, and engineering.
