Problem 28 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand the expression. $$ \left(2 A+B^{2}\right)^{4} $$

Step-by-Step Solution

Answer

The expansion is \(16A^4 + 32A^3B^2 + 24A^2B^4 + 8AB^6 + B^8\).

Step 1: Understanding the Binomial Theorem

The Binomial Theorem helps in expanding expressions of the form \((x + y)^n\). It states:\[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}y^k\]where \(\binom{n}{k}\) is the binomial coefficient.

Step 2: Identify 'x', 'y', and 'n'

In the expression \((2A + B^2)^4\), identify 'x', 'y', and 'n':- \(x = 2A\)- \(y = B^2\)- \(n = 4\)

Step 3: Calculate the Binomial Coefficients

Calculate the binomial coefficients \(\binom{4}{k}\) for each term from 0 to 4:\[\begin{align*}\binom{4}{0} &= 1, \\binom{4}{1} &= 4, \\binom{4}{2} &= 6, \\binom{4}{3} &= 4, \\binom{4}{4} &= 1.\end{align*}\]

Step 4: Apply the Binomial Theorem

Now, expand the expression using the Binomial Theorem:\[(2A + B^2)^4 = \sum_{k=0}^{4} \binom{4}{k} (2A)^{4-k} (B^2)^k\]

Step 5: Calculate Each Term of the Expansion

Compute each term:- For \(k=0\): \[\binom{4}{0}(2A)^4(B^2)^0 = 1 \cdot 16A^4 \cdot 1 = 16A^4\]- For \(k=1\): \[\binom{4}{1}(2A)^3(B^2)^1 = 4 \cdot 8A^3 \cdot B^2 = 32A^3B^2\]- For \(k=2\): \[\binom{4}{2}(2A)^2(B^2)^2 = 6 \cdot 4A^2 \cdot B^4 = 24A^2B^4\]- For \(k=3\): \[\binom{4}{3}(2A)^1(B^2)^3 = 4 \cdot 2A \cdot B^6 = 8AB^6\]- For \(k=4\): \[\binom{4}{4}(2A)^0(B^2)^4 = 1 \cdot 1 \cdot B^8 = B^8\]

Step 6: Combine Terms for Final Expansion

Combine all the terms to form the final expanded expression:\[16A^4 + 32A^3B^2 + 24A^2B^4 + 8AB^6 + B^8\]

Key Concepts

Understanding Binomial Expansion

The concept of binomial expansion is useful whenever we want to expand expressions raised to a power. Imagine you have an equation like \((x + y)^n\). The binomial expansion allows us to break it down into individual terms. Think of it like opening a box of numbers. The main idea is to represent it as a sum of different terms involving powers of \(x\) and \(y\). Each term is built using the binomial coefficients and the individual powers of \(x\) and \(y\).

This helps when, instead of multiplying everything out manually, you can follow a systematic approach to find out what each term should look like. The result is a neatly expanded version of the equation, saving you a lot of work while also having the potential to reveal interesting patterns in between terms.

Exploring Binomial Coefficients

Binomial coefficients are the magic numbers that help determine the weight each term carries in a binomial expansion. If we're dealing with \((x + y)^n\), the binomial coefficient for each term is represented as \(\binom{n}{k}\). It tells us how many ways we can choose \(k\) items from a total of \(n\) without regard to the order. To find this value, we use the formula:

• \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

The exclamation mark "!" indicates a factorial, which means you multiply the number by every whole number less than itself (e.g., \(4! = 4 \times 3 \times 2 \times 1\)).

In the context of binomial expansion, these coefficients play a crucial role as multipliers for each term. They ensure that we count each combination of terms correctly. When you apply them in your problem-solving, make sure to compute all the coefficients for your range of \(k\), from 0 to \(n\). This provides you the structure necessary for each summed term in the expanded expression.

Polynomial Expressions and their Expansion

Polynomial expressions are mathematical expressions consisting of variables raised to various powers, multiplied by coefficients. These expressions, like \((2A + B^2)^4\) in our example, are central to algebra, often showing up in different areas of mathematics and applied fields. When we expand these using the binomial theorem, we are breaking it down into a sum of simpler monomial terms.

Each term in the polynomial expansion reflects a unique combination of the variables and their powers. The understanding of polynomials and their behavior is fundamental, as they often represent real-world problems like calculating areas, predicting outcomes, or even modeling trajectories. Remember, expanding this involves identifying not just the terms, but how each variable interacts with others through multiplication, often leading to striking insights upon simplification.

Once expanded, polynomial expressions can be analyzed for various properties, including finding roots or solving quadratic equations. This makes mastering their expansion critical for higher-level math and beyond.