Animated Solution for Mathematics - Binomial Theorem: The term independent of $x$ in the expression of $(1-x^2+3x^3){\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11}$, $x\ne 0$ is

We are tasked with finding the term independent of $x$ in the expression $(1-x^2+3x^3){\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11}$. When you see a polynomial multiplied by a binomial power, do not panic.

Instead, view it as a three-pronged strategy. We are looking for the coefficient of $x^0$. Because we have three terms in the leading polynomial, we have three potential ways to reach that $x^0$ target.

We cannot expand the binomial ${\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11}$ fully; that would be inefficient and error-prone. Instead, we use the General Term formula. For any binomial $(a+b{)}^n$, the general term is $T_{r+1}={}^n{C}_r{a}^{n-r}{b}^r$.

Applying this to our specific binomial, we get:

Now, let us isolate the variables. By grouping the constants and the powers of $x$, we simplify this to:

Simplifying the exponent of $x$, we get $33-3r-2r$, which is $33-5r$. Thus, the general term is:

Now, we distribute the leading polynomial $(1-x^2+3x^3)$ across our general term. This gives us three distinct scenarios to check for the $x^0$ term:

1. The $1$ term: We need $x^{33-5r}=x^0$, so $33-5r=0$. This gives $r=6.6$, which is not an integer. No solution exists here.

2. The $-x^2$ term: We need $-x^2\cdot x^{33-5r}=x^0$. This implies $x^{2+33-5r}=x^0$, or $35-5r=0$. Solving this gives $r=7$, which is a valid integer.

3. The $3x^3$ term: We need $3x^3\cdot x^{33-5r}=x^0$. This implies $x^{3+33-5r}=x^0$, or $36-5r=0$. This gives $r=7.2$, which is not an integer. No solution exists here.

We have found our winner: $r=7$. Now, we plug this back into our expression to find the coefficient:

Using the symmetry property, ${}^{11}{C}_7={}^{11}{C}_4=330$. Substituting this in, we get:

The two negatives cancel out to a positive. Simplifying the powers of $5$, we are left with:

Reducing this fraction, we arrive at the final result: