Question

\(10^{\text {th }}\) term in expansion of \(\left[2 x^{2}+(1 / x)\right]^{12}\) is \(\ldots \ldots \ldots\) (a) \(\left[(1760) / \mathrm{x}^{2}\right]\) (b) \(\left[(1760) / \mathrm{x}^{3}\right]\) (c) \(\left[(880) / \mathrm{x}^{2}\right]\) (d) \(\left[(880) / \mathrm{x}^{3}\right]\)

Short answer

The 10th term in the expansion of \(\left[2 x^{2}+(1 / x)\right]^{12}\) is \(\frac{1760}{x^3}\).

Step-by-step solution

Identify the general term formula for binomial expansion

For finding the 10th term in the expansion, we'll use the general term formula in binomial expansion, which is given by \(T_{r+1} = \binom{n}{r} a^{n-r} b^{r}\) where r is the index of the term starting from 0. In our case, we want the 10th term, so \(r = 9\). #Step 2: Substitute the values of a, b, n, and r in the formula#

Substituting the given values

Now, we will substitute the given values into our formula. - a = 2x² - b = 1/x - n = 12 - r = 9 \(T_{10} = \binom{12}{9} (2x^2)^{12-9} (1/x)^9 = \binom{12}{9} (2x^2)^3 (1/x)^9\) #Step 3: Calculate the binomial coefficient and simplify#

Calculate the binomial coefficient and simplify the terms

Now, we'll evaluate the binomial coefficient \(\binom{12}{9}\) and simplify the terms. \(\binom{12}{9}\) = \(\frac{12!}{9!3!}\) = \(\frac{12\times 11\times 10}{3\times 2\times 1}\) = 220 Moreover, \((2x^2)^3 = 8x^6\) \((1/x)^9 = 1/x^9\) #Step 4: Calculate the 10th term#

Calculate the 10th term

Combine the results from Step 3 to find the value of the 10th term \(T_{10} = 220 * 8x^6 * 1/x^9 = 1760/x^3\) So, the 10th term in the expansion is \(\frac{1760}{x^3}\), which corresponds to option (b).

Key concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows us to expand expressions raised to a power without directly multiplying the expression by itself repeatedly. When you have an expression of the form \( (a + b)^n \), where \( n \) is a positive integer, the binomial theorem provides a systematic method for expanding it into a sum involving terms of the form \( a^{n-r}b^r \).

Each term in the expansion is accompanied by a binomial coefficient, denoted by \( \binom{n}{r} \), which represents the number of ways to choose \( r \) elements from a set of \( n \) elements. The binomial theorem is expressed as \[ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r}b^r \. \]
This theorem saves us from the tedium of manual expansion and provides a direct route to any specific term in the polynomial sequence. To find the 10th term in our example, we used the binomial theorem to identify the general form of any term in the expansion as \( T_{r+1} = \binom{n}{r} a^{n-r} b^{r} \). Understanding and applying this theorem is the key to solving many algebraic problems that involve polynomial expansions.

Binomial Coefficient

At the heart of each term in a binomial expansion lies the binomial coefficient. This element of the expression provides the number of combinations or ways in which an event can occur without considering the order. In the language of mathematics, it's often represented as \( \binom{n}{r} \) which is read as 'n choose r'. The formula for calculating a binomial coefficient is \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \. \]

This means that to find the binomial coefficient for our \( 10^{th} \) term example, we compute \( \binom{12}{9} \) by evaluating \( \frac{12!}{9!3!} \) which simplifies to \( \frac{12 \times 11 \times 10}{3 \times 2 \times 1} \) giving us 220. The concept of binomial coefficients is not only vital in algebra but also extends to various fields such as statistics, probability theory, and combinatorics. Through understanding binomial coefficients, students gain insight into the structure of polynomial terms and how they are weighted.

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial expression that has been raised to a power as a series of terms added together. Each term in the expanded polynomial will consist of variables and coefficients. The coefficients can sometimes be determined using combination methods, such as the binomial coefficients when the polynomial is a binomial.

The example we are dissecting involves expanding the polynomial \( (2x^2 + \frac{1}{x})^{12} \) to find a specific term. By using the general term formula of binomial expansion, we calculated the 10th term directly without having to expand the entire binomial. In general, \( a^n \), a polynomial of degree one raised to the power of \( n \) can be expanded directly, but for polynomials of degrees higher than one, we employ structures like the binomial theorem to manage the complexity.

Application in Our Exercise

In the exercise provided, we applied the principles of polynomial expansion to find that the 10th term is \( \frac{1760}{x^3} \) by organizing our work into recognizable polynomial terms and then combining like terms. This illustrates the practical utility of polynomial expansion in breaking down high-power expressions into manageable pieces.