(1%2Bx%5E2)(1%2Bx%5E3)(1%2Bx%5E4).jpeg "(1+x)(1+x^2)(1+x^3)(1+x^4)")
Expanding the Product (1+x)(1+x^2)(1+x^3)(1+x^4)
This article will explore the expansion of the product (1+x)(1+x^2)(1+x^3)(1+x^4). We will use the distributive property and discuss patterns that emerge in the expansion.
Expanding the Product Step-by-Step
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Start with the first two factors: (1+x)(1+x^2) = 1 + x^2 + x + x^3 = 1 + x + x^2 + x^3
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Multiply the result by the third factor: (1 + x + x^2 + x^3)(1+x^3) = 1 + x^3 + x + x^4 + x^2 + x^5 + x^3 + x^6 = 1 + x + x^2 + 2x^3 + x^4 + x^5 + x^6
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Finally, multiply the result by the fourth factor: (1 + x + x^2 + 2x^3 + x^4 + x^5 + x^6)(1+x^4) = 1 + x^4 + x + x^5 + x^2 + x^6 + 2x^3 + 2x^7 + x^4 + x^8 + x^5 + x^9 + x^6 + x^10 = 1 + x + x^2 + 2x^3 + 2x^4 + 2x^5 + 2x^6 + 2x^7 + x^8 + x^9 + x^10
Observations and Patterns
- Coefficients: Notice that most of the coefficients are 1 or 2. This is because each term is generated by multiplying a combination of 'x' terms from the original factors.
- Exponents: The exponents in the final expansion range from 0 to 10. This makes sense as the highest power of x in the original product is x^4 * x^3 * x^2 * x = x^10.
Generalization
The expansion of the product (1+x)(1+x^2)(1+x^3)...(1+x^n) will have the following characteristics:
- Exponents: The exponents will range from 0 to the sum of the exponents in the original factors.
- Coefficients: The coefficients will be determined by the number of ways to combine 'x' terms from the original factors to get the desired exponent.
While calculating the exact coefficients for a large 'n' can be complex, understanding these patterns can help simplify the process.