%5E3-expand-formula.jpeg "(x-1)^3 Expand Formula")
Expanding the Formula: (x-1)^3
In algebra, expanding formulae is an essential skill to master. One of the most common formulas that students encounter is the binomial expansion of (x-1)^3. In this article, we will explore the expansion of this formula and its applications.
What is Binomial Expansion?
Binomial expansion is a mathematical operation that involves expanding an expression of the form (a+b)^n, where a and b are variables and n is a positive integer. The result of the expansion is a polynomial with terms that involve various combinations of a and b, raised to different powers.
The Formula: (x-1)^3
The formula (x-1)^3 is a special case of binomial expansion, where a = x and b = -1. To expand this formula, we can use the binomial theorem, which states that:
(a+b)^n = a^n + na^(n-1)b + n(n-1)a^(n-2)b^2 + … + b^n
In our case, we have a = x and b = -1, so we can substitute these values into the formula:
(x-1)^3 = x^3 + 3x^2(-1) + 3x(-1)^2 + (-1)^3
Expanding the Formula
Now, let's expand the formula by simplifying each term:
x^3: This term remains unchanged, as x is raised to the power of 3.
3x^2(-1): This term becomes -3x^2, as we multiply x^2 by -1.
3x(-1)^2: This term becomes 3x, as (-1)^2 equals 1.
(-1)^3: This term becomes -1, as (-1) is raised to the power of 3.
Final Expansion
By combining these simplified terms, we get the final expansion of the formula:
(x-1)^3 = x^3 - 3x^2 + 3x - 1
This expanded formula is a polynomial with four terms, each involving x raised to a different power.
Applications of (x-1)^3
The formula (x-1)^3 has several applications in mathematics and computer science, including:
Algebraic Manipulation
The expanded formula can be used to simplify complex algebraic expressions involving x and constants.
Geometric Progressions
The formula can be used to calculate the sum of geometric progressions with a common ratio of -1.
Computer Science
The formula is used in algorithms for solving certain types of recurrence relations.
Conclusion
In this article, we have explored the expansion of the formula (x-1)^3 using the binomial theorem. We have derived the final expansion and discussed its applications in various fields. Mastering this formula is essential for students of mathematics and computer science, as it provides a powerful tool for solving complex problems.
