Question

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{0}\) in the expansion of \(\left(x^{2}+\frac{1}{x}\right)^{12}\)

Short answer

495

Step-by-step solution

Identify the General Term

Identify the general term in the binomial expansion of ewline ewline ewline ewline ewline ewline ewline ewline \[(x^{2}+\frac{1}{x})^{12}\].The general term of ewline ewline ewline ewline ewline ewline ewline \((x^{2}+\frac{1}{x})^{12}\) ewline ewline ewline ewline is given by ewline ewline ewline ewline \[{T}_{k+1} = \binom{12}{k} (x^{2})^{12-k} (\frac{1}{x})^{k}.\]

Simplify the General Term

Simplify the general term expression: ewline ewline ewline ewline ewline ewline ewline ewline\({T}_{k+1} = \binom{12}{k} x^{2(12-k)} x^{-k} = \binom{12}{k} x^{24-3k}.\)

Set the Exponent to Zero

Find the value of ewline ewline ewline ewline ewline ewline ewline \(k\) such that the exponent of x is zero: ewline ewline ewline ewline \(24-3k = 0\). ewline ewline ewline ewline Now, solve for ewline ewline ewline ewline \(k\): ewline ewline ewline ewline\(3k = 24\). ewline ewline ewline ewline ewline ewline ewline ewline ewline \(k = 8\).

Substitute ewline ewline ewline ewline \(k\) ewline ewline ewline ewline to Find the Coefficient

Substitute ewline ewline ewline ewline \(k = 8\) ewline ewline ewline ewline back into the general term ewline ewline ewline ewline to get the coefficient: ewline ewline ewline ewline ewline \({T}_{9} = \binom{12}{8} = \binom{12}{4} = 495.\) Hence, the coefficient of ewline ewline ewline ewline \(x^0\) ewline ewline ewline ewline in the expansion is ewline ewline ewline ewline 495.

Key concepts

Binomial Expansion

Let's start with the binomial expansion. This method helps us expand expressions of the form \((a + b)^n\). The binomial theorem states that: \[ (a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \text{...} + \binom{n}{n} a^0 b^n \] To find a specific term in the expansion, we use the formula for the general term: \[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \] Here, \(\binom{n}{k}\) represents the binomial coefficient, which we'll discuss next. This formula is essential because it helps us find any term without expanding the whole expression. Using this, you can pinpoint the term you need, like the coefficient of \(x^0\)

Coefficients

Coefficients play a crucial role in any algebraic expansion. In the binomial theorem, they are represented using binomial coefficients. The binomial coefficient is calculated as: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] Here, \( n! \) (n factorial) means multiplying all whole numbers from \( n \) down to 1. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). These coefficients help us weigh the terms in the binomial expansion. They tell us how many times each term is repeated, which is vital when calculating the specific term's value. For instance, in our solution, when we substitute \( k = 8 \), we compute \( \binom{12}{8} = 495 \).

Algebra

Algebra serves as the foundation for understanding binomial expansions and coefficients. It involves using letters and symbols to represent numbers and quantities in formulas and equations.
Let’s break it down further:

• \textbf{Variables:} Symbols (like x) used to represent unknown values.
• \textbf{Expressions:} Combinations of numbers and variables (like \(x^2 + \frac{1}{x}\)).
• \textbf{Equations:} Statements that two expressions are equal (like \(24 - 3k = 0\)).

When working with algebra, we often need to simplify expressions or solve equations. In our problem, we started with the expression \((x^2+\frac{1}{x})^{12}\). We then simplified the general term to find where the exponent of x is zero, leading us to solve for \(k\). By combining these elements, algebra lets us break down and solve complex problems step-by-step.