Question

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{0}\) in the expansion of \(\left(x-\frac{1}{x^{2}}\right)^{9}\)

Short answer

The coefficient of \(x^{0}\) is -84.

Step-by-step solution

Identify the relevant term

To find the coefficient of a specific term in a binomial expansion, identify which term in the expansion matches the criteria. We are looking for the coefficient of the term where the power of x is 0.

Express the general term using Binomial Theorem

The Binomial Theorem states that \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\] Here, let’s set \(a = x\), \(b = -\frac{1}{x^2}\), and \(n = 9\). The general term of the expansion is given by: \[T_k = \binom{9}{k} x^{9-k} \left( -\frac{1}{x^2} \right)^k\]

Simplify the general term

Simplify the general term \(T_k\): \[T_k = \binom{9}{k} x^{9-k} \left(-1\right)^k \left( \frac{1}{x^2} \right)^k = \binom{9}{k} (-1)^k x^{9-k-2k} = \binom{9}{k} (-1)^k x^{9-3k}\]

Find the term where power of x is 0

Set the power of x in the term equal to 0: \[9 - 3k = 0\] Solve for k: \[3k = 9 \ k = 3\] This means that the term with \(k=3\) will have \(x^0\), which is a constant.

Calculate the coefficient

Substitute \(k=3\) back into the simplified general term to find the coefficient: \[T_3 = \binom{9}{3} (-1)^3 x^0 = \binom{9}{3} (-1)^3\] Calculate \(\binom{9}{3}\): \[\binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9!}{3!6!} = 84\] Therefore, the coefficient is \[84(-1)^3 = -84\]

Key concepts

Binomial Expansion

Binomial expansion is a method used to expand expressions that are raised to a power. According to the Binomial Theorem, any binomial raised to a power can be expressed as a sum of terms involving the coefficients, the binomial terms, and their respective powers. This statement is formalized as:
\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
Here, \(a\) and \(b\) are the terms of the binomial, and \(n\) is the exponent to which the binomial is raised. The terms in the expansion are represented by \(\binom{n}{k} a^{n-k} b^k\). The Binomial Theorem is widely used in algebra for simplifying polynomials and solving algebraic equations.

Coefficients

Coefficients are the numerical factors in algebraic expressions that multiply variables or powers of variables. In the context of the Binomial Theorem, coefficients are given by the binomial coefficients, represented as \(\binom{n}{k}\). Each term in the binomial expansion has a specific coefficient, which can be calculated based on \(n\) and \(k\), where \(n\) is the exponent of the binomial and \(k\) is the term index. To find the coefficient of a specific term, identify the relevant \(k\) value and calculate the binomial coefficient. For example, in the expansion of \((x - \-\frac{1}{x^2})^9\), identifying \(k = 3\) lets us find the coefficient of \(-84\) in the \(x^0\) term.

Combinatorial Coefficients

Combinatorial coefficients, also known as binomial coefficients, represent the number of ways to choose \(k\) elements from a set of \(n\) elements. These coefficients are essential in calculating the terms of a binomial expansion and are denoted as \(\binom{n}{k}\). The formula to compute these coefficients is:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
Here, \(n! (n\) factorial) represents the product of all positive integers up to \(n\), and similarly for \(k!\) and \( (n-k)!\). In the example provided, calculating \(\binom{9}{3}\) involves:
\[ \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9!}{3!6!} = 84 \]
These coefficients help us understand the distribution of terms in polynomial expansions.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. For instance, \(x^{-2}\) can be written as \(\frac{1}{x^2}\). In binomial expansions, negative exponents often require careful handling to ensure correct simplification. For example, in the expression \((x - \frac{1}{x^2})^9\), substituting \(b = -\frac{1}{x^2}\) and expanding it using the Binomial Theorem results in terms containing negative exponents. These terms are then simplified into a form involving positive exponents when required. Proper algebraic manipulation ensures the correctness of the expansion and the terms that match the required exponents are identified efficiently.