%5E3-formula-class-9.jpeg "(a+b+c)^3 Formula Class 9")
Cube of the Sum of Three Terms: (a+b+c)^3 Formula for Class 9
In algebra, one of the most important formulas to learn is the cube of the sum of three terms, denoted by (a+b+c)^3. This formula is widely used in various mathematical problems, including algebra, geometry, and trigonometry. In this article, we will discuss the formula, its proof, and some examples to illustrate its application.
The Formula:
The cube of the sum of three terms (a+b+c)^3 can be expanded as:
(a+b+c)^3 = a^3 + b^3 + c^3 + 3a^2(b+c) + 3b^2(a+c) + 3c^2(a+b) + 6abc
This formula can be used to expand the cube of any three terms, whether they are numbers, variables, or expressions.
Proof:
The proof of the formula involves multiplying the expression (a+b+c) three times using the distributive property of multiplication over addition. Here's a brief outline of the proof:
- (a+b+c) × (a+b+c) = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
- (a^2 + b^2 + c^2 + 2ab + 2bc + 2ca) × (a+b+c) = a^3 + b^3 + c^3 + 3a^2(b+c) + 3b^2(a+c) + 3c^2(a+b) + 6abc
Simplifying the expression, we get the final formula:
(a+b+c)^3 = a^3 + b^3 + c^3 + 3a^2(b+c) + 3b^2(a+c) + 3c^2(a+b) + 6abc
Examples:
Let's apply the formula to some examples:
Example 1:
Expand (2x+3y+z)^3 using the formula.
(2x+3y+z)^3 = (2x)^3 + (3y)^3 + z^3 + 3(2x)^2(3y+z) + 3(3y)^2(2x+z) + 3z^2(2x+3y) + 6(2x)(3y)(z)
Simplifying the expression, we get:
(2x+3y+z)^3 = 8x^3 + 27y^3 + z^3 + 36x^2y + 54xy^2 + 12xz^2 + 18yz^2 + 72xyz
Example 2:
Expand (a+2b+3c)^3 using the formula.
(a+2b+3c)^3 = a^3 + (2b)^3 + (3c)^3 + 3a^2(2b+3c) + 3(2b)^2(a+3c) + 3(3c)^2(a+2b) + 6a(2b)(3c)
Simplifying the expression, we get:
(a+2b+3c)^3 = a^3 + 8b^3 + 27c^3 + 6a^2b + 12ab^2 + 18a^2c + 27ac^2 + 36b^2c + 54bc^2 + 72abc
Conclusion:
In this article, we have learned the formula for the cube of the sum of three terms, (a+b+c)^3. We have also seen how to apply the formula to different examples. This formula is an essential tool in algebra and can be used to solve various mathematical problems.
