Problem 21 - College Algebra with Modeling and Visualization

Question

Use the binomial theorem to expand each expression. $$ (2 x-3)^{3} $$

Step-by-Step Solution

Answer

\((2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27\).

Step 1: Identify Components

The binomial theorem states that \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\] For this exercise, identify that \(a = 2x\), \(b = -3\), and \(n = 3\).

Step 2: Expand Using the Theorem

Substitute the values into the binomial theorem formula. The expansion becomes:\[(2x - 3)^3 = \sum_{k=0}^{3} \binom{3}{k} (2x)^{3-k} (-3)^k\]

Step 3: Calculate Each Term

Now calculate each term from the sum:- For \(k=0\): \[\binom{3}{0} (2x)^{3} (-3)^{0} = 1 \, \cdot \, 8x^3 \, \cdot \, 1 = 8x^3\]- For \(k=1\): \[\binom{3}{1} (2x)^{2} (-3)^{1} = 3 \, \cdot \, 4x^2 \, \cdot \, (-3) = -36x^2\]- For \(k=2\): \[\binom{3}{2} (2x)^{1} (-3)^{2} = 3 \, \cdot \, 2x \, \cdot \, 9 = 54x\]- For \(k=3\): \[\binom{3}{3} (2x)^{0} (-3)^{3} = 1 \, \cdot \, 1 \, \cdot \, (-27) = -27\]

Step 4: Combine the Terms

Combine all the calculated terms to get the expanded expression:\[(2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27\]

Key Concepts

Binomial Expansion

The binomial expansion allows us to express algebraic expressions raised to a power as a sum of terms. It's a fundamental concept used in algebra to simplify the computation of expressions that involve binomials. When you work with a binomial raised to a power, like \((a+b)^n\), you don't need to multiply it out manually. Instead, you can use the binomial theorem to quickly find the expanded form.
It's based on the pattern \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where each term consists of the binomial coefficient \(\binom{n}{k}\), multiplied by \(a\) to the power of \((n-k)\) and \(b\) to the power of \(k\).
This gives us a structured way to expand binomials using:

• Binomial coefficients to determine the number of ways specific terms can appear.
• Powers of individual terms as determined by their positions.

Algebraic Expressions

Understanding algebraic expressions like \((2x - 3)^3\) is important for performing operations such as binomial expansion. Algebraic expressions are composed of variables, numbers, and arithmetic operations. In this case, our expression consists of two terms, \(2x\) and \(-3\), which form the binomial to be expanded.
To effectively expand algebraic expressions using the binomial theorem, you should:

• Identify the parts of the binomial, i.e., what represents \(a\) and what represents \(b\).
• Determine the exponent \(n\), which tells you how many terms your expansion will include.

By breaking down an algebraic expression into its components, you can then apply the binomial theorem efficiently.
This approach helps in simplifying calculations and understanding the structure of expansions, turning complex expressions into simpler, more manageable sums.

Combinatorial Coefficients

Combinatorial coefficients, also known as binomial coefficients, are crucial in binomial expansion. These numbers appear in Pascal's triangle and reflect the coefficients in front of each term when you expand a binomial. For an expression \((a+b)^n\), the binomial coefficient \(\binom{n}{k}\) determines how many times the product \(a^{n-k}b^k\) appears in the expansion.
Mathematically, these coefficients are defined as \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(!\) denotes factorial, which is the product of a number with all positive integers less than itself.
To compute these quickly:

• Recognize that each row in Pascal's triangle corresponds to the powers of \((a+b)^n\).
• For any given \(n\), the coefficients start from 1, build up to a peak, then mirror back down to 1.

Using combinatorial coefficients not only streamlines computations but also reveals interesting patterns, offering insight into the symmetries inherent in algebra.