Animated Solution for Mathematics - Binomial Theorem: The absolute difference of the coefficients of $x^{10}$ and $x^7$ in the expansion of $(2x^2+\frac{1}{2x}{)}^{11}$ is equal to

Imagine you are standing before a massive, complex expression: $(2x^2+\frac{1}{2x}{)}^{11}$. It looks intimidating, but the Binomial Theorem is your master key.

It allows us to peek into the structure of this expansion without ever having to write it all out. We are going to find the coefficients of $x^{10}$ and $x^7$ with surgical precision.

The General Term

The heart of the Binomial Theorem is the general term formula:

Think of this as a machine. In our case, $n=11$, $a=2x^2$, and $b=\frac{1}{2x}$.

When we plug these into our machine, we get:

The Algebraic Cleanup

Now, let's perform some algebraic surgery to separate the constants from the variables. We distribute the powers: $(2x^2{)}^{11-r}$ becomes $2^{11-r}\cdot x^{22-2r}$, and $(\frac{1}{2x}{)}^r$ becomes $2^{-r}\cdot x^{-r}$.

When we combine these, we get:

Simplifying the powers of $2$ and $x$ gives us the elegant form:

The Hunt for the Coefficients

We want the coefficient of $x^{10}$. We set the exponent $22-3r=10$, which leads to $3r=12$, or $r=4$.

The coefficient is:

Calculating this, $\left(\genfrac{}{}{0ex}{}{11}{4}\right)=330$ and $2^3=8$. Thus, $330\times 8=2640$.

Now, let's repeat this for $x^7$. We set $22-3r=7$, which gives $3r=15$, so $r=5$.

The coefficient is:

Calculating this, $\left(\genfrac{}{}{0ex}{}{11}{5}\right)=462$ and $2^1=2$. Thus, $462\times 2=924$.

The Final Victory

We have our two coefficients: $2640$ and $924$. The problem asks for the absolute difference:

We have successfully navigated the complexity of the binomial expansion. The result matches the pattern $12^3-12=1728-12=1716$.