Finding the Coefficient of x¹¹ in a Binomial Expansion

We need to find the coefficient of the \(x^{11}\) term in the expansion of \({\left( {{x^2} -\frac{1}{x}} \right)^{10}}\). This involves using the binomial theorem.

The general term in the binomial expansion of \({\left( {a + b} \right)^n}\) is given by the formula:

\(T_{r+1} = \binom{n}{r} a^{n-r} b^r\)

In our case, the expression is \({\left( {{x^2} -\frac{1}{x}} \right)^{10}}\). Comparing this to \({\left( {a + b} \right)^n}\), we have:

• \(n = 10\)
• \(a = x^2\)
• \(b = -\frac{1}{x} = -x^{-1}\)

Now, let's write the general term, \(T_{r+1}\), for the given expansion:

\(T_{r+1} = \binom{10}{r} \left(x^2\right)^{10-r} \left(-x^{-1}\right)^r\)

Let's simplify the powers of x in the term:

• \(\left(x^2\right)^{10-r} = x^{2 \times (10-r)} = x^{20-2r}\)
• \(\left(-x^{-1}\right)^r = (-1)^r (x^{-1})^r = (-1)^r x^{-r}\)

Substitute these back into the general term formula:

\(T_{r+1} = \binom{10}{r} x^{20-2r} (-1)^r x^{-r}\)

Combine the terms with x by adding their exponents:

\(T_{r+1} = \binom{10}{r} (-1)^r x^{20-2r - r}\)

\(T_{r+1} = \binom{10}{r} (-1)^r x^{20-3r}\)

The coefficient of this term is \( \binom{10}{r} (-1)^r \), and the power of x is \( x^{20-3r} \). We are looking for the coefficient of \(x^{11}\). So, we need to set the exponent of x equal to 11:

\(20 - 3r = 11\)

Now, we solve for r:

\(20 - 11 = 3r\)

\(9 = 3r\)

\(r = \frac{9}{3}\)

\(r = 3\)

So, the term containing \(x^{11}\) is the term where \(r=3\), which is the \(T_{3+1} = T_4\) term. The coefficient of this term is given by \( \binom{10}{r} (-1)^r \) with \(r=3\):

Coefficient \( = \binom{10}{3} (-1)^3\)

First, let's calculate the binomial coefficient \( \binom{10}{3} \). This is calculated as:

\(\binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10!}{3!7!}\)

\(\binom{10}{3} = \frac{10 \times 9 \times 8 \times 7!}{ (3 \times 2 \times 1) \times 7! }\)

Cancel out the 7! term:

\(\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1}\)

\(\binom{10}{3} = \frac{720}{6}\)

\(\binom{10}{3} = 120\)

Next, calculate the \( (-1)^3 \) part:

\((-1)^3 = -1\)

Now, multiply these two parts to get the final coefficient of the \(x^{11}\) term:

Coefficient \( = \binom{10}{3} (-1)^3 = 120 \times (-1) = -120\)

The coefficient of \(x^{11}\) in the binomial expansion is -120.

The calculation for the binomial expansion coefficient \( \binom{10}{3} \) resulted in 120.

This step-by-step process using the binomial theorem helps determine the specific **term in expansion** and its associated coefficient.

The final coefficient is -120.