Problem 56
Question
Find the middle term in the expansion of $$\left(\frac{1}{x}-x^{2}\right)^{12}$$.
Step-by-Step Solution
Verified Answer
The middle term of the expansion of \(\left(\frac{1}{x}-x^{2}\right)^{12}\) is -924x.
1Step 1: Identifying the elements
First identify elements 'a' and 'b' in function \(\left(\frac{1}{x}-x^{2}\right)^{12}\). Where 'a' is \(\frac{1}{x}\) and 'b' is -\(x^{2}\). Also, as the exponent of the function is 12, the number of terms in the expansion will be 12+1 = 13
2Step 2: Finding the middle term
The 7th term is the middle term of the expansion as 13 is an odd number. Using the formula for the rth term of a binomial expansion which is \({n \choose r-1} a^{n-r+1}b^{r-1}\), where 'n' is the power of the function, 'a' and 'b' are the two terms and 'r' is the term number, substitute the corresponding values. So, the 7th term = \({12 \choose 7-1} (\frac{1}{x})^{12-7+1}(-x^{2})^{7-1}\)
3Step 3: Calculating the 7th term
After substituting the values, simplify the expression for the 7th term. Perform the binomial coefficient calculation, subtract exponents and combine like terms. This simplifies to -924x.
Key Concepts
Binomial TheoremCoefficientsAlgebraic Expressions
Binomial Theorem
The binomial theorem is a fundamental principle in algebra that expands expressions raised to a power. This theorem allows expressions of the form \((a + b)^n\) to be expanded into a series of terms. Each term is expressed in the manner of products of 'a' and 'b' raised to various powers.
For example, if you have \((a + b)^n\), the binomial expansion will include terms like \({n \choose k}a^{n-k}b^k\), where \({n \choose k}\) is the binomial coefficient, determining the number of ways to choose 'k' elements from 'n' items.
The full expansion includes terms from \((a^n)b^0\) to \((a^0)b^n\). Each increment in 'k' reduces the power of 'a' by one and increases the power of 'b' by one, until all combinations are exhausted.
The binomial theorem is particularly useful for expanding polynomial expressions and is an essential tool in combinatorics and probability theory. Understanding it allows students to simplify and solve complex algebraic problems.
For example, if you have \((a + b)^n\), the binomial expansion will include terms like \({n \choose k}a^{n-k}b^k\), where \({n \choose k}\) is the binomial coefficient, determining the number of ways to choose 'k' elements from 'n' items.
The full expansion includes terms from \((a^n)b^0\) to \((a^0)b^n\). Each increment in 'k' reduces the power of 'a' by one and increases the power of 'b' by one, until all combinations are exhausted.
The binomial theorem is particularly useful for expanding polynomial expressions and is an essential tool in combinatorics and probability theory. Understanding it allows students to simplify and solve complex algebraic problems.
Coefficients
Coefficients in the context of binomial expansion are numerical factors that accompany terms in a polynomial expression. They result from the combination of items, represented through binomial coefficients.
In the formula \({n \choose k}a^{n-k}b^k\), \({n \choose k}\) is the coefficient of the particular term k. These coefficients obey Pascal's Triangle, where each number is the sum of the two directly above it.
In the formula \({n \choose k}a^{n-k}b^k\), \({n \choose k}\) is the coefficient of the particular term k. These coefficients obey Pascal's Triangle, where each number is the sum of the two directly above it.
- The central element of Pascal's Triangle simplifies the computation of binomial coefficients, providing an intuitive pattern.
- For example, to find a coefficient with \(n = 12\) and \(k = 6\), find the value in the 12th row and the 7th column. This value is '924'. Hence, the coefficient for the 7th term is 924.
Algebraic Expressions
Algebraic expressions involve the arrangement of numbers, variables, and operations into meaningful constructs. The exercise showcases a binomial expression, which is a specific type with two distinct terms.
In the problem \((\frac{1}{x} - x^2)^{12}\), you have 'a' as \(\frac{1}{x}\) and 'b' as \(-x^2\). Applying the binomial theorem breaks it into recognizable terms and allows us to manipulate and interpret it systematically.
Simplifying these expressions involves performing operations like addition, subtraction, multiplication, division, and exponentiation on these variables and constants.
In the problem \((\frac{1}{x} - x^2)^{12}\), you have 'a' as \(\frac{1}{x}\) and 'b' as \(-x^2\). Applying the binomial theorem breaks it into recognizable terms and allows us to manipulate and interpret it systematically.
Simplifying these expressions involves performing operations like addition, subtraction, multiplication, division, and exponentiation on these variables and constants.
- Rewriting components such as \(x^n\) or \(\frac{1}{x^m}\) ensures the expression aligns with the powers expected within a structured polynomial.
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