Find the expanded and simplified form of \((x+2x^2)^6\) using the binomial theorem.

Options:
A. \(x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64\)
B. \(x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1\)
C. \(x^6 + 10x^5 + 40x^4 + 80x^3 + 120x^2 + 160x + 256\)

Given the binomial theorem, the expanded form of (x+2x^2)^6 will be derived from taking each term of the binomial, raising it to a power that decreases from 6 to 0, and multiplying by the appropriate binomial coefficient found in Pascal's triangle.

Explanation:

The question asks for the expanded and simplified form of (x+2x2)6 using the binomial theorem. The binomial theorem states (a + b)n = an+nan-1b and so on. By applying this axiom, none of the provided options fit. However, to construct our own solution, each term of the expansion would be formed by taking a term from the binomial, raising it to a power that decreases from 6 to 0, and then multiplying by the appropriate binomial coefficient. This process is repeated for all terms of the binomial. The coefficients or the numbers in front of each term can be found in Pascal's triangle under the 6th row.

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Find the expanded and simplified form of \((x+2x^2)^6\) using the binomial theorem. Options: A. \(x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64\) B. \(x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1\) C. \(x^6 + 10x^5 + 40x^4 + 80x^3 + 120x^2 + 160x + 256\)

Answer :

Final answer:

Explanation:

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Other Questions

Find the expanded and simplified form of \((x+2x^2)^6\) using the binomial theorem.

Options:
A. \(x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64\)
B. \(x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1\)
C. \(x^6 + 10x^5 + 40x^4 + 80x^3 + 120x^2 + 160x + 256\)