Expansion of 12x - x^2^6 up to x^3

In this article, we will explore the expansion of the polynomial 12x - x^2^6 up to the x^3 term using the binomial theorem. The binomial theorem is a powerful mathematical tool that allows us to expand expressions of the form a b raised to a power.

Understanding the Binomial Theorem

The binomial theorem states:

a b^n sum_{k0}^{n} binom{n}{k} a^{n-k} b^k

Here, binom{n}{k} represents the binomial coefficient, which can be calculated as binom{n}{k} n! / (k!(n-k)!). In our case, we will consider a 1 and b 2x - x^2, and we will focus on the terms up to x^3.

Step-by-step Expansion

Step 1: Expansion for k 0

Term for k 0: binom{6}{0} 1^6 (2x - x^2)^0 1

Step 2: Expansion for k 1

Term for k 1: binom{6}{1} 1^5 (2x - x^2)^1 6(2x - x^2) 12x - 6x^2

Step 3: Expansion of (2x - x^2)^2

First, expand (2x - x^2)^2: (2x - x^2)^2 4x^2 - 4x^3 x^4

Now, contribute the terms up to x^3:15(4x^2 - 4x^3) 6^2 - 6^3

Step 4: Expansion for k 2

Term for k 2: binom{6}{2} 1^4 (2x - x^2)^2 15(2x - x^2)^2 15(4x^2 - 4x^3 x^4) 6^2 - 6^3 15x^4

We only need the terms up to x^3, so we disregards 15x^4.

Step 5: Expansion of (2x - x^2)^3

First, expand (2x - x^2)^3: (2x - x^2)^3 8x^3 - 12x^4 6x^5 - x^6

Now, contribute the terms up to x^3: 20(8x^3 - 12x^4 6x^5 - x^6) 16^3 - 24^4 12^5 - 2^6

We only need the terms up to x^3, so we disregard 24^4, 12^5, and 2^6.

Combining All Relevant Terms

Combining all the relevant terms from the above expansions, we get:

From ( k 0 ): 1 From ( k 1 ): 12x - 6x^2 From ( k 2 ): 6^2 - 6^3 15x^4 From ( k 3 ): 16^3 - 24^4 12^5 - 2^6

Summing these up, we get:

1 12x - 6x^2 6^2 - 6^3 16^3

This simplifies to:

1 12x 54x^2 10^3

Conclusion

Thus, the expansion of ( 12x - x^2^6 ) up to ( x^3 ) is:

1 12x 54x^2 10^3

Additional Expansion Techniques

Additionally, there are alternative methods for expanding and simplifying the polynomial. For instance, we can cube the polynomial first and then square the result, or square the polynomial first and then cube it. Both methods will eventually lead to the same result.

For instance:

Cubing First: 1(x^2 - x^3)^3 x^6 - 3x^5 3x^4 - x^3

Squaring First: 1(2x - x^2)^2 4x^2 - 4x^3 x^4

By considering only the terms up to ( x^3 ) in both cases, we can simplify the polynomial and arrive at the same result.