Finding the Term Independent of x in Binomial Expansion Without Full Expansion

When working with binomial expansions, it's often useful to find the term that is independent of x without fully expanding the expression. This process can be streamlined using the binomial theorem, which is a powerful tool in algebra. This article will illustrate this methodology with two examples: 2x - frac{3}{x^2}^9 and 2x^2 - frac{1}{9}^9, demonstrating how to find the desired term without extensive calculation.

1. Binomial Theorem and Expansion

The binomial theorem states that:

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

This theorem will be used to simplify the process of finding the term independent of x in the given expansions.

2. Example 1: Finding the Term Independent of x in (2x - frac{3}{x^2})^9

Let (a 2x) and (b -frac{3}{x^2}), with (n 9). The general term in the expansion is given by:

(T_k binom{9}{k} (2x)^{9-k} left(-frac{3}{x^2}right)^k)

After simplification, this becomes:

(T_k binom{9}{k} 2^{9-k} x^{9-k} (-3)^k x^{-2k})

Combining the powers of x gives:

(T_k binom{9}{k} 2^{9-k} (-3)^k x^{9-3k})

To find the term independent of x, set the exponent of x to zero:

9 - 3k 0 quad Rightarrow quad 3k 9 quad Rightarrow quad k 3)

Substituting (k 3) back into the expression for (T_k) gives:

(T_3 binom{9}{3} 2^{9-3} (-3)^3)

Calculating each part:

(binom{9}{3} frac{9 times 8 times 7}{3 times 2 times 1} 84)

(2^6 64)

((-3)^3 -27)

Thus, the term independent of x is:

(T_3 84 times 64 times -27 -84 times 64 times 27)

Therefore, the term independent of x in the expansion of 2x - frac{3}{x^2}^9 is:

(-84 times 64 times 27)

3. Example 2: Finding the Term Independent of x in (2x^2 - frac{1}{9})^9

Using the binomial theorem and the properties of exponents, the term independent of x in the expansion of 2x^2 - frac{1}{9}^9 can be found as:

(binom{9}{9} cdot (2x^2)^{9-9} left(-frac{1}{9}right)^9 -frac{1}{9^9})

Thus, the term independent of x in the expansion of 2x^2 - frac{1}{9}^9 is:

-frac{1}{9^9} approx -frac{1}{387,420,489})

Conclusion

By using the binomial theorem and the properties of exponents, we can efficiently find the term independent of x in a binomial expansion without fully expanding the expression. This method is particularly useful in simplifying complex algebraic problems and reducing computational effort.

References

[1] Binomial Theorem - Wikipedia

[2] Binomial Theorem - MathIsFun