Key Concept: Binomial Theorem and General Term The Binomial Theorem provides a formula for expanding expressions of the form $(a+b)^n$. For any non-negative integer $n$, the general term (or the $(r+1)^{th}$ term) in the expansion of $(a+b)^n$ is given by: $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ where $\binom{n}{r} = \frac{n!}{r!(n-r)!}$ is the binomial coefficient. This formula allows us to find any specific term or its coefficient without performing the entire expansion.

Step 1: Identify the Components of the Given Expression The given expression is $( 2 x^3 - \frac{1}{3 x^2} )^5$. To apply the general term formula, we need to identify $a$, $b$, and $n$ from this expression.

• $a = 2x^3$
• $b = -\frac{1}{3x^2}$ (Note: It's crucial to include the negative sign with the second term)
• $n = 5$

Step 2: Apply the General Term Formula and Simplify Substitute these identified values into the general term formula $T_{r+1} = \binom{n}{r} a^{n-r} b^r$: $T_{r+1} = \binom{5}{r} (2x^3)^{5-r} \left(-\frac{1}{3x^2}\right)^r$ Now, we simplify this expression by separating the numerical coefficients and the powers of $x$. Remember the exponent rules: $(xy)^m = x^m y^m$, $(x^p)^q = x^{pq}$, and $\frac{1}{x^k} = x^{-k}$. $T_{r+1} = \binom{5}{r} (2)^{5-r} (x^3)^{5-r} \left(-\frac{1}{3}\right)^r \left(\frac{1}{x^2}\right)^r$ $T_{r+1} = \binom{5}{r} (2)^{5-r} x^{3(5-r)} \left(-\frac{1}{3}\right)^r x^{-2r}$ Next, we combine the terms involving $x$ by adding their exponents: $T_{r+1} = \binom{5}{r} (2)^{5-r} \left(-\frac{1}{3}\right)^r x^{15-3r-2r}$ $T_{r+1} = \binom{5}{r} (2)^{5-r} \left(-\frac{1}{3}\right)^r x^{15-5r}$ This simplified general term now clearly separates the numerical coefficient part from the variable part $x^{15-5r}$.

Step 3: Determine the Value of 'r' for the Coefficient of $x^5$ We are asked to find the coefficient of $x^5$. This means the exponent of $x$ in our general term, which is $15-5r$, must be equal to 5. Set the exponent equal to 5 and solve for $r$: $15 - 5r = 5$ Subtract 15 from both sides: $-5r = 5 - 15$ $-5r = -10$ Divide by -5: $r = \frac{-10}{-5}$ $r = 2$ The value of $r$ is 2. It is essential to check that $r$ is a non-negative integer and $0 \le r \le n$. Here, $0 \le 2 \le 5$, which is a valid value for $r$.

Step 4: Calculate the Coefficient Now that we have $r=2$, we substitute this value back into the coefficient part of the general term (the expression without $x^{15-5r}$): $\text{Coefficient} = \binom{5}{2} (2)^{5-2} \left(-\frac{1}{3}\right)^2$ Let's calculate each part:

• Binomial Coefficient: $\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(3 \times 2 \times 1)} = \frac{120}{12} = 10$
• First Power Term: $(2)^{5-2} = (2)^3 = 8$
• Second Power Term: $\left(-\frac{1}{3}\right)^2 = \left(\frac{-1}{3}\right) \times \left(\frac{-1}{3}\right) = \frac{1}{9}$

Finally, multiply these results to get the full coefficient: $\text{Coefficient} = 10 \times 8 \times \frac{1}{9}$ $\text{Coefficient} = \frac{80}{9}$

Tips and Common Mistakes to Avoid

• Sign Errors: Always be careful with negative signs in the terms, especially when they are raised to a power. An even exponent makes a negative base positive, while an odd exponent keeps it negative.
• Exponent Laws: Ensure correct application of exponent rules like $(x^a)^b = x^{ab}$ and $1/x^a = x^{-a}$ to combine variable terms accurately. A common error is incorrectly adding exponents instead of multiplying or mismanaging reciprocals.
• Validity of 'r': After calculating $r$, always verify that it is a non-negative integer and falls within the range $0 \le r \le n$. If $r$ is not an integer or is outside this range, it implies that the desired term does not exist in the expansion, and its coefficient is 0.
• Arithmetic Precision: Perform calculations step-by-step, particularly with fractions, to minimize arithmetic errors.

Summary and Key Takeaway To efficiently find the coefficient of a specific power of $x$ in a binomial expansion, the most effective method is to use the general term formula. This involves:

1. Correctly identifying the components $a$, $b$, and $n$.
2. Setting up the general term $T_{r+1} = \binom{n}{r} a^{n-r} b^r$.
3. Systematically simplifying the general term to isolate the exponent of $x$.
4. Equating this exponent to the target power of $x$ to solve for $r$.
5. Substituting the valid $r$ back into the numerical part of the general term to compute the final coefficient. Following these steps ensures a precise and accurate solution. The coefficient of $x^5$ in the given expansion is $\frac{80}{9}$.