%5E3-class-9.jpeg "(x+y+z)^3 Class 9")
Cubing a Trinomial: (x+y+z)^3 for Class 9
In algebra, cubing a trinomial is an essential concept that involves expanding the cube of a sum of three variables. In this article, we will explore the formula and process for cubing a trinomial, specifically (x+y+z)^3, which is an important topic for Class 9 students.
The Formula:
The formula for cubing a trinomial is given by:
(x+y+z)^3 = x^3 + y^3 + z^3 + 3x^2(y+z) + 3y^2(x+z) + 3z^2(x+y) + 6xyz
This formula might look complex, but it can be easily understood by breaking it down into smaller parts.
Breaking Down the Formula:
Let's analyze the formula step by step:
- x^3 + y^3 + z^3: These are the cubes of individual variables.
- 3x^2(y+z): This term is obtained by multiplying x^2 with the sum of y and z.
- 3y^2(x+z): This term is obtained by multiplying y^2 with the sum of x and z.
- 3z^2(x+y): This term is obtained by multiplying z^2 with the sum of x and y.
- 6xyz: This term is obtained by multiplying x, y, and z.
Simplifying the Expansion:
Now that we have broken down the formula, let's simplify the expansion of (x+y+z)^3:
(x+y+z)^3 = x^3 + y^3 + z^3 + 3x^2y + 3x^2z + 3y^2x + 3y^2z + 3z^2x + 3z^2y + 6xyz
By rearranging the terms, we get:
(x+y+z)^3 = x^3 + y^3 + z^3 + 3xy(x+y) + 3yz(y+z) + 3zx(z+x) + 6xyz
This is the simplified expansion of (x+y+z)^3.
Examples and Practice:
Now that we have understood the formula and process for cubing a trinomial, let's practice it with some examples:
Example 1: Expand (2x+3y+z)^3
Solution: Using the formula, we get:
(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)
Example 2: Expand (x+2y+3z)^3
Solution: Using the formula, we get:
(x+2y+3z)^3 = x^3 + (2y)^3 + (3z)^3 + 3x^2(2y+3z) + 3(2y)^2(x+3z) + 3(3z)^2(x+2y) + 6x(2y)(3z)
Conclusion:
In this article, we have discussed the formula and process for cubing a trinomial, specifically (x+y+z)^3. We have broken down the formula, simplified the expansion, and practiced it with examples. This concept is crucial for Class 9 students to understand and master algebraic expressions.
