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Chapter 10: Problem 56

Find the middle term in the expansion of \(\left(\frac{1}{x}-x^{2}\right)^{12}\)

Short Answer

Expert verified
The middle term in the expansion of \( \left(\frac{1}{x}-x^{2}\right)^{12} \) is -924x^3.

Step by step solution

01

Understand the Exponent

The given expression to expand is \(\left(\frac{1}{x}-x^{2}\right)^{12}\). The exponent here is 12, an even number. When the exponent is an even number, the middle term is given by \(\frac {n}{2} + 1\), where \(n\) is the exponent. So, in this case, the middle term will be the \(\frac{12}{2} + 1 = 7th\) term.
02

Apply Binomial Theorem

The general form of the \(r^{th}\) term in the expansion of \( (a+b)^n \) by Binomial theorem is \( T_{r} = ^nC_{r-1}\cdot a^{n-r+1} \cdot b^{r-1} \). Apply this formula to find the 7-th term. Let \( a = \frac{1}{x} \) and \( b = -x^2 \), and \( n = 12 \). The 7-th term will be \( T_7 = ^{12}C_{6}\cdot (\frac{1}{x})^{12-6+1} \cdot (-x^2)^{6-1} \).
03

Simplify

Simplify the expression to find its value. Here, \( ^{12}C_{6} = 924 \), and \(\frac{1}{x^{12-6+1}}\) can be rewritten as \( \frac{1}{x^7} \), and \((-x^2)^{6-1}\) can be rewritten as \( -x^{10} \). So, the 7-th term becomes \( T_7 = 924 \cdot \frac{1}{x^7} \cdot -x^{10} = -924x^3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expansion of Expressions
In mathematics, expanding expressions is a method of expressing a mathematical equation in an extended form. When you expand an expression, you're transforming it from a compact form into a longer, more detailed version. For example, the expression \((a+b)^n\) can be expanded using known algebraic rules.
This process is handy in simplifying complex equations, solving algebraic problems, and helping to find specific terms within the expanded form.
  • In our exercise, the expression \(\left( \frac{1}{x} - x^2 \right)^{12}\) needs to be expanded to identify certain terms, like the middle term.
  • The Binomial Theorem is the tool we use to perform such expansions efficiently and accurately.
Mastering expansion techniques sharpens algebra skills, enabling you to solve a wide range of problems with ease.
Middle Term in Binomial Expansion
Finding the middle term in a binomial expansion is a common requirement in algebra. When the exponent of the expression is even, determining the middle term is straightforward using a formula.
The middle term can be found at the position \(\frac{n}{2} + 1\) for an even exponent \(n\).
  • In our example \(\left( \frac{1}{x} - x^2 \right)^{12}\), the exponent is 12, which is even. Hence, the middle term is at the 7th position \(\left(\frac{12}{2} + 1\right)\).
  • This 7th term is significant as it represents the centerpiece of the symmetrical expansion.
Knowing how to locate this middle term allows deeper insights into the properties and behavior of the expanded polynomial.
Combinatorics in Algebra
Combinatorics plays a pivotal role in algebra, especially when dealing with binomial expansions. It involves counting, arranging, and choosing elements - all crucial in deriving specific terms in an expression.
In the binomial expansion \((a+b)^n\), the coefficient of each term is determined using combinatorial numbers, known as binomial coefficients, which are symbolized as \( ^nC_r \).
  • In our problem, finding the 7th term specifically requires calculating \( ^{12}C_6 \), which equals 924.
  • These coefficients are central for expanding and simplifying expressions into desired terms.
Understanding combinatorial concepts such as arrangers and coefficients can enhance algebraic problem-solving and offers a structured approach to calculating terms in expansions.

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Most popular questions from this chapter

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