%5E3-formula-expansion.jpeg "(a+b+c)^3 Formula Expansion")
Expanding (a + b + c)^3
The expansion of (a + b + c)^3 can be a tedious process if done manually. However, there's a handy formula to make it much easier. Here's how to understand and apply it:
The Formula
The formula for expanding (a + b + c)^3 is:
(a + b + c)^3 = a^3 + b^3 + c^3 + 3a^2b + 3a^2c + 3ab^2 + 3ac^2 + 3b^2c + 3bc^2 + 6abc
Understanding the Formula
- Cubic Terms: You start with the cubes of each individual term (a^3, b^3, c^3).
- Square Terms: Then, you get terms where each variable is squared and multiplied by the other two variables (3a^2b, 3a^2c, 3ab^2, 3ac^2, 3b^2c, 3bc^2). Notice the coefficient of 3 for each of these terms.
- Mixed Term: Finally, you have a term where all three variables are multiplied together, with a coefficient of 6 (6abc).
Example
Let's expand (x + y + z)^3 using the formula:
(x + y + z)^3 = x^3 + y^3 + z^3 + 3x^2y + 3x^2z + 3xy^2 + 3xz^2 + 3y^2z + 3yz^2 + 6xyz
Applying the Formula
To use the formula, simply replace 'a', 'b', and 'c' with the respective terms in your expression. For instance:
- (2x + 3y + z)^3: Replace 'a' with 2x, 'b' with 3y, and 'c' with z.
Key Points
- Symmetry: Notice that the formula is symmetrical. You can rearrange the variables without affecting the result.
- Combinations: The coefficients in the formula represent the number of ways you can choose the variables in each term. For example, the coefficient of 3 for the square terms represents the three ways you can choose two variables out of three.
- Pascal's Triangle: You can also use Pascal's triangle to find the coefficients of the expanded terms.
Conclusion
Understanding the formula for expanding (a + b + c)^3 simplifies the process significantly. It allows you to quickly and accurately expand expressions without resorting to tedious manual multiplication. Remember to apply the formula carefully and pay attention to the coefficients and the order of the variables.