%5E2.jpeg "(x+x^2+x^3+x^4+x^5+x^6)^2")
Expanding the Square of a Polynomial: (x+x^2+x^3+x^4+x^5+x^6)^2
This article explores the process of expanding the square of the polynomial (x+x^2+x^3+x^4+x^5+x^6)^2. We'll break down the steps and present the final result.
Understanding the Problem
We are tasked with squaring the polynomial (x+x^2+x^3+x^4+x^5+x^6). This means multiplying the polynomial by itself:
(x+x^2+x^3+x^4+x^5+x^6) * (x+x^2+x^3+x^4+x^5+x^6)
Expanding the Expression
To expand this, we can use the distributive property. However, doing this directly would be tedious. Instead, we can employ a helpful pattern:
- Recognize the pattern: The polynomial is a geometric series where each term is 'x' times the previous term.
- Use a formula: The sum of a finite geometric series is given by: S = a(1-r^n)/(1-r) where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
Step 1: Find the sum of the geometric series
In our case, a = x, r = x, and n = 6. Substituting these values into the formula:
S = x(1-x^6)/(1-x)
Step 2: Square the sum
Now, we need to square the sum we just found:
(x(1-x^6)/(1-x))^2
Step 3: Simplify
Simplifying the expression gives us:
x^2 (1 - x^6)^2 / (1 - x)^2
Step 4: Expand the numerator
Expanding the numerator, we get:
x^2 (1 - 2x^6 + x^12) / (1 - x)^2
Step 5: Final Result
Therefore, the expansion of (x+x^2+x^3+x^4+x^5+x^6)^2 is:
(x^2 - 2x^8 + x^14) / (1 - x)^2
Conclusion
This method demonstrates a more efficient approach to expanding the square of the given polynomial by utilizing the properties of geometric series. The final result provides the expanded form of the expression in a simplified format.