Problem 57 - Precalculus with Limits

Question

In Exercises 53 - 60, find the coefficient of the term in the expansion of the binomial. Binomial \( \quad \quad \quad \) Term \( \left(2x - 5y\right)^{9} \quad \quad \quad ax^4y^5 \)

Step-by-Step Solution

Answer

The coefficient of the term \(x^4y^5\) in the expansion of the binomial \((2x - 5y)^{9}\) is \(-6300000\).

Step 1: Identify the Parameters

In the given binomial \((2x - 5y)^9\), \(a = 2x\), \(b = -5y\) and \(n = 9\). In the term \(ax^4*y^5\), \(k = 5\). Plug these values into the Binomial Theorem, remembering to adjust for the negative value of \(b\).

Step 2: Use the Binomial Theorem to Find the Coefficient

According to the Binomial Theorem, the term \(ax^4y^5\) in the expansion of the binomial is given by \(C(n,k) * a^{n-k} * b^k\). Substituting our parameters, this becomes, \(C(9,5) * (2x)^{9-5} * (-5y)^5\). Calculate the value of this expression to get the coefficient \(a\).

Step 3: Calculate the Coefficient

Start by calculating the binomial coefficient \(C(9,5)\), which is equal to 9! / [(9-5)! * 5!]. This equals 126. Next, calculate \( (2x)^{9-5}\) which is \(16x^4\). Finally, calculate \((-5y)^5\), which equals -3125y^5. Multiply these results together: (126) * (16x^4) * (-3125y^5) simplifies to \(-6300000x^4y^5\). The coefficient \(a\) of the term \(x^4y^5\) is thus -6300000.

Key Concepts

Binomial Expansion

The binomial expansion is a powerful algebraic tool for expanding expressions of the form \((a + b)^n\).
This means you take a binomial, which is an expression with two terms, and expand it into a polynomial with multiple terms.
Each term in the expanded form is a product of powers of the original terms and a specific binomial coefficient. To begin the expansion process:

• Identify the terms in the binomial, \(a\) and \(b\), along with the exponent \(n\).
• Use the Binomial Theorem, which states that \((a + b)^n = \sum_{k=0}^n C(n, k) a^{n-k} b^k\).
• Here, \(C(n, k)\) represents the binomial coefficient for each term.

The Binomial Theorem helps in not only expanding the expression but also in identifying specific terms within the expansion.
If you are interested in a particular term, notice its powers of \(a\) and \(b\), which guides you to the correct term during the expansion.

Binomial Coefficient

The binomial coefficient, often denoted as \(C(n, k)\), is a key part of a binomial expansion.
It is the coefficient of the term \(a^{n-k}b^k\) in the expansion of \((a + b)^n\).
The binomial coefficient formula is given by \[ C(n, k) = \frac{n!}{k!(n-k)!} \]where \(!\) represents factorial, which is the product of all positive integers up to that number. For example:

• \(n! = n \times (n-1) \times (n-2) \times \, \cdots \, \times 1\)

Binomial coefficients are symmetrical, meaning \(C(n, k) = C(n, n-k)\), which can simplify calculations.
They not only appear in binomial expansions but also in combinatorial problems, representing ways to choose \(k\) elements from a set of \(n\) elements.
Understanding how to calculate and apply these coefficients is crucial for working with binomials.

Polynomial Expansion

Polynomial expansion converts a binomial raised to an exponent into a longer expression with several terms.
Each term involves powers of the original elements of the binomial and a specific binomial coefficient.
For example, expanding \((a + b)^n\) yields a polynomial with \(n+1\) terms if fully expanded.When you expand a polynomial:

• Begin with the highest power of the first term and gradually reduce its exponent in each subsequent term.
• Concurrently, increase the power of the second term starting from zero.
• The sum of the exponents of both terms in any particular term of the expansion will always equal \(n\).

This systematic method ensures no terms are missed and helps simplify complex expressions.
While expanding, using organized methods like the Binomial Theorem makes the process efficient and error-free.
The result is a full expression showcasing the relationship between coefficients, variable powers, and factorial calculations.