Problem 52 - Algebra and Trigonometry

Question

In the expansion of \(\left[a+(b+c)^{2}\right]^{8}\) find the coefficient of the term containing \(a^{5} b^{4} c^{2}\).

Step-by-Step Solution

Answer

The coefficient is 336.

Step 1: Simplify the Inner Expression

First, simplify the inner expression \(b+c\)^{2}\: \[ (b+c)^{2} = b^{2} + 2bc + c^{2} \]

Step 2: Expand the Outer Expression

Expand the given expression \[\left[a+(b+c)^{2}\right]^{8} \] using the binomial theorem: \[ \sum_{k=0}^{8} \binom{8}{k} a^{8-k} (b^{2} + 2bc + c^{2})^{k} \] We need to find the coefficient of the term containing \ a^{5} b^{4} c^{2} \.

Step 3: Expand the Inner Binomial

Consider the term \(b^{2} + 2bc + c^{2})^{k}\) and expand it again using the binomial theorem: \[ \sum_{i=0}^{k} \binom{k}{i} b^{2(k-i)} (2bc)^{i} c^{2(k-i)} \]

Step 4: Identify the Relevant Terms

For the term \a^{5} b^{4} c^{2}\, set \(8-k) = 5\) to solve for \k: \ k = 3\. Then expand \ (b^{2} + 2bc + c^{2})^{3} \: \[ \sum_{i=0}^{3} \binom{3}{i} b^{2(3-i)} (2bc)^{i} c^{2(3-i)} \] We need the term with \b^{4} c^{2}\.

Step 5: Combine for Correct Terms

From the binomial expansion, find the appropriate combination of terms such that \(i=1\): \[ \binom{3}{1} b^{2(3-1)} (2bc)^{1} c^{2(3-1)} = 3 b^{4} 2bc c^{2} = 6 b^{4} c^{2} \]

Step 6: Determine the Coefficient

Combine all necessary coefficients: \[ \binom{8}{3} \binom{3}{1} \cdot 6= 56 \cdot 6 = 336 \] Thus, the coefficient of \a^{5} b^{4} c^{2}\ is 336.

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows us to expand expressions raised to a power.
It states that for any integers \(a\), \(b\), and \(n\), the expansion of \((a + b)^n\) is: \[\binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \text{...} + \binom{n}{n}b^n \]
Each term in the expansion is made up of a binomial coefficient \(\binom{n}{k}\), a power of \(a\), and a power of \(b\).
The binomial coefficient \(\binom{n}{k}\) is calculated using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
For example, in the given exercise, we first expanded \((b+c)^2\) to simplify the inner part of the original expression.

Coefficient Identification

Identifying the coefficient of a specific term in a polynomial expansion is crucial.
Let's walk through the process as explained in the exercise.
First, expand to find a general form for all terms. Then, locate the term containing the exact powers we need.
In this case, we seek the coefficient of \(a^5 b^4 c^2\) in the expanded form of \((a+(b+c)^2)^8\).
After using the binomial theorem several times, we identify the terms that include the required powers.
For example, setting \(8-k = 5\) to find \(k = 3\), and then expanding \((b^2 + 2bc + c^2)^3\) we identified the term \(\binom{3}{1}b^4(2bc)c^{2} = 6 b^4 c^2\).

Polynomial Expansion

Polynomial expansion refers to expressing a polynomial raised to a power as a sum of terms.
Each term consists of different combinations of the original variables raised to different powers.
Using the binomial theorem, we expanded \((a + (b+c)^2)^8\): \[\binom{8}{k}a^{8-k}(b^2 + 2bc + c^2)^k \]
As demonstrated, further expanding each \((b^2 + 2bc + c^2)^k\) into its binomial components provides all individual terms.
The goal is to systematically combine the variables until identifying the exact required term.
Steps involve multiple expansions and careful attention to the powers of each variable.

Algebraic Expressions

Algebraic expressions include variables, coefficients, and constants combined using arithmetic operations.
Understanding and manipulating these expressions are foundational skills for solving equations.
In the exercise, we worked with expressions like \(a + (b+c)^2\) and expanded them step-by-step.
Simplifying within the parentheses first ensures that complex expressions transform into simpler polynomial forms.
Each manipulation follows algebraic rules, particularly those governing exponents and coefficients.
Mastering these manipulations is essential for working through problems involving complex expressions.
It’s not just about expanding terms but about strategically breaking down components to ultimately find what’s needed.