(a+b+c)3 Formula Solution

4 min read Jul 03, 2024

(a+b+c)³ Formula Solution

The formula for (a+b+c)³ is a well-known algebraic identity that is widely used in various mathematical calculations. In this article, we will explore the solution to this formula and its applications.

What is (a+b+c)³ Formula?

The (a+b+c)³ formula is an algebraic identity that represents the cube of the sum of three variables a, b, and c. It is expressed as:

(a+b+c)³ = a³ + b³ + c³ + 3a²(b+c) + 3b²(a+c) + 3c²(a+b) + 6abc

This formula can be derived using various methods, including the binomial theorem and the method of induction.

Proof of (a+b+c)³ Formula

One way to prove the (a+b+c)³ formula is to use the binomial theorem, which states that:

(a+b)³ = a³ + 3a²b + 3ab² + b³

Using this theorem, we can write:

(a+b+c)³ = ((a+b) + c)³

Expanding the right-hand side using the binomial theorem, we get:

(a+b+c)³ = (a+b)³ + 3(a+b)²c + 3(a+b)c² + c³

Expanding the (a+b)² and (a+b)c² terms, we get:

(a+b+c)³ = a³ + 3a²b + 3ab² + b³ + 3(a²c + 2abc + bc²) + 3(ac² + b²c) + c³

Simplifying the expression, we get:

(a+b+c)³ = a³ + b³ + c³ + 3a²(b+c) + 3b²(a+c) + 3c²(a+b) + 6abc

This is the required formula.

Applications of (a+b+c)³ Formula

The (a+b+c)³ formula has various applications in mathematics and other fields, including:

Algebraic Manipulations: The formula is used to simplify complex algebraic expressions and to expand the cube of a sum of variables.
Geometry: The formula is used to calculate the volume of a rectangular prism and other geometric shapes.
Trigonometry: The formula is used to simplify trigonometric identities and to solve trigonometric equations.
Physics: The formula is used to calculate the kinetic energy of an object and to solve problems in mechanics.

Conclusion

In this article, we have derived and proved the (a+b+c)³ formula using the binomial theorem. We have also discussed the various applications of the formula in mathematics and other fields. The formula is an important tool in algebraic manipulations and is widely used in various mathematical calculations.