Problem 47 - Beginning and Intermediate Algebra

Question

Use the binomial theorem to expand each expression. $$\left(\frac{1}{2} m-3 n\right)^{4}$$

Step-by-Step Solution

Answer

Using the binomial theorem, we expand the given expression as follows: \(\left(\frac{1}{2} m-3 n\right)^{4} = \frac{1}{16}m^4 - 6m^3n + 54m^2n^2 - 324mn^3 + 81n^4\).

Step 1: Identify a, b, and n

We have \(a = \frac{1}{2}m\), \(b = -3n\), and \(n = 4\).

Step 2: Set up the summation for the binomial theorem

According to the binomial theorem, the expansion of \((a+b)^n\) is given by the summation: \[\sum_{k=0}^n {n \choose k} a^{n-k} b^k\] We need to evaluate this for our specific values, which are \(a = \frac{1}{2}m\), \(b = -3n\), and \(n = 4\). The expansion of the expression will be: \[\sum_{k=0}^4 {4 \choose k} \left(\frac{1}{2}m\right)^{4-k}(-3n)^{k}\]

Step 3: Expand the summation term by term

Now, we can expand the summation term by term: k = 0: $${4 \choose 0}\left(\frac{1}{2}m\right)^4(-3n)^0 = \frac{1}{16}m^4$$ k = 1: $${4 \choose 1}\left(\frac{1}{2}m\right)^3(-3n)^1 = -6m^3n$$ k = 2: $${4 \choose 2}\left(\frac{1}{2}m\right)^2(-3n)^2 = 54 m^2 n^2$$ k = 3: $${4 \choose 3}\left(\frac{1}{2}m\right)^1(-3n)^3 = -324 m n^3$$ k = 4: $${4 \choose 4}\left(\frac{1}{2}m\right)^0(-3n)^4 = 81 n^4$$

Step 4: Combine the terms

Now, we can combine the terms in the expansion: \[\frac{1}{16}m^4 - 6m^3n + 54m^2n^2 - 324mn^3 + 81n^4\] So the expanded expression using the binomial theorem is: \[\left(\frac{1}{2} m-3 n\right)^{4} = \frac{1}{16}m^4 - 6m^3n + 54m^2n^2 - 324mn^3 + 81n^4\]

Key Concepts

Polynomial Expansion

Polynomial expansion is a process where you take a power expression and rewrite it as a sum of terms. Each term is a product of coefficients and variables raised to various powers. This process is especially handy in algebra and calculus because it allows you to simplify expressions and solve equations more easily.

The Binomial Theorem is particularly useful for expanding polynomial expressions of the form \((a + b)^n\). The theorem provides a straightforward way to calculate each term in the expanded form. When we speak of "expansion," we mean taking a single expression and rewriting it as a series of terms which are easier to handle or analyze.

In our exercise, we're expanding the expression \(\left(\frac{1}{2}m - 3n\right)^4\) using the binomial theorem. This means that we're reconfiguring it from a power form into a polynomial with multiple terms, each representing different permutations of the initial variables raised to powers.

Binomial Coefficients

In any polynomial expansion using the binomial theorem, binomial coefficients play a crucial role. They are the numbers that multiply the variables in each term of the expanded polynomial. These coefficients are derived from the combination formula \({n \choose k}\), where \(n\) is the power to which the binomial is raised, and \(k\) represents each successive term.

Here are a few important points about binomial coefficients:

• They are calculated as \({n \choose k} = \frac{n!}{k!(n-k)!}\).
• The coefficients form a symmetric pattern; for example, \({n \choose k} = {n \choose n-k}\).
• Pascals' Triangle visually represents these coefficients and is a handy tool for finding them quickly.
  
  

In the given exercise, our power \(n\) is 4. Thus, the sequence of coefficients when expanding \(\left(\frac{1}{2}m - 3n\right)^4\) involves calculating \({4 \choose k}\) for each term from \(k = 0\) to \(k = 4\). This process results in the coefficients 1, 4, 6, 4, and 1, respectively, which are then used to multiply the respective terms of the expansion.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, division, and exponentiation). They form the basis of algebra, allowing us to represent complex relationships and solve problems across mathematics and science fields.

Understanding algebraic expressions requires recognizing the components within them, such as:

• Variables, which stand for unknown values and can change within the scope of the expression.
• Constants, which are fixed values.
• Operators, which indicate actions like addition (\(+\)) or multiplication (\(\times\)).
  
  

In expanding algebraic expressions using the binomial theorem, we identify each part of the expression separately. In the context of our exercise, the terms \(\frac{1}{2}m\) and \(-3n\) are our algebraic expressions within the binomial. Once expanded, these parts combine with binomial coefficients to form a larger expression, illustrating how algebraic manipulation leads to new insights and solutions.