Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the coefficient \(a\) of the given term in the expansion of the binomial. Binomial =\((x-2 y)^{10}\) Term = \(a x^{8} y^{2}\)

Step-by-Step Solution

Answer

The coefficient of the term \(x^{8} y^{2}\) in the expansion of \((x-2 y)^{10}\) is 180.

Step 1: Understand the binomial theorem and its coefficients

According to the binomial theorem, the coefficient of \(x^p y^q\) in the expansion of \((x+y)^n\) is given by \(\binom{n}{p}\), if \(p+q=n\). In this case, \(n=10\), \(p=8\), and \(q=2\), so the formula can be applied.

Step 2: Apply the binomial theorem

We can apply the binomial theorem, plugging in \(n=10\), \(p=8\), and \(q=2\), letting \(a = \binom{10}{8} = 45\). This only considers the \(x^{8} y^{2}\) term though and thus isn't the full answer, as we also need to consider the multiplier that stems from the given binomial equation.

Step 3: Consider the multiplier in the original equation

In the given binomial, \(x-2y\), note that the y-term is not simply \(y\), but expanded as \(-2y\), thus it inversely affects the expression on expansion. In the term being considered, \(x^{8} y^{2}\), there are \(y^{2}\). So, the multiplier effect would be \((-2)^{2} = 4\).

Step 4: Multiply to get the final answer

The coefficient \(a\) thus becomes the product of the binomial coefficient and multiplier: \(a=45*4 = 180\).

Key Concepts

Binomial Coefficient

The binomial coefficient is a key element in the binomial theorem. It helps us find specific terms in a polynomial expansion. The binomial coefficient is notated as \( \binom{n}{k} \), read as "n choose k". This represents the number of ways to choose \(k\) elements from a total of \(n\) elements without regard to order.

• It is calculated using the formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \(!\) denotes a factorial.
• For example, \( \binom{10}{8} = \frac{10!}{8!2!} \).

In the problem provided, we calculated \( \binom{10}{8} = 45 \). This coefficient tells us how many ways we can select terms to form \( x^8 y^2 \) in the expansion \((x-2y)^{10}\).
The binomial coefficient is an essential part of identifying the desired term's coefficient before considering other variables like multiplication factors from the equation.

Polynomial Expansion

Expanding a polynomial like \((x-2y)^{10}\) involves finding each term in the expanded form. This process is simplified by the binomial theorem, which lets us break down the expansion without multiplying everything manually.

• The binomial theorem gives each term as \( \binom{n}{k} x^{n-k} y^{k} \).
• For example, in our problem, this becomes \( \binom{10}{8}(x)^{10-8}(-2y)^{8} \).

We are interested in the \( x^8 y^2 \) term. By the theorem, the coefficient \( \binom{10}{8} \) and the adjustments for powers and signs come from the binomial's \(-2y\) factor.
In polynomial expansions, capturing the effect of coefficients like \(-2\) is crucial for accuracy, dictating how terms expand beyond their direct binomial coefficients.

Pascal's Triangle

Pascal's Triangle is a visual representation of binomial coefficients arranged in a triangular fashion. Each number is the sum of the two numbers directly above it. This pattern helps us quickly find coefficients when expanding binomials.

• The nth row gives us the coefficients for \((x+y)^{n}\).
• For an expansion like \((x-2y)^{10}\), the 10th row provides coefficients for each term.

For example, the 10th row is: 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, mirroring the coefficients of \((x+y)^n\).
When using Pascal's Triangle, we're quickly able to spot the 45 used in our problem, representing the \( x^8 y^2 \) term. While the Triangle simplifies coefficient identification, note that other transformations (like from \(-2y\)) must be applied separately.