Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated operations and simplify. \((2 t+3)^{3}\)

Step-by-Step Solution

Answer

The expression \((2t+3)^3\) simplifies to \(8t^3 + 36t^2 + 54t + 27\).

Step 1: Identify the Binomial Formula

To expand the expression \((2t+3)^3\), we will use the binomial theorem. The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = 2t\), \(b = 3\), and \(n = 3\).

Step 2: Calculate the Binomial Coefficients

For \(n = 3\), calculate the binomial coefficients using \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). The coefficients will be:\[\binom{3}{0} = 1, \ \binom{3}{1} = 3, \ \binom{3}{2} = 3, \ \binom{3}{3} = 1.\]

Step 3: Apply the Binomial Formula

Substitute the values of \(a\), \(b\), and the calculated coefficients into the binomial formula:\[\begin{align*}(2t+3)^3 &= \binom{3}{0}(2t)^3(3)^0 + \binom{3}{1}(2t)^2(3)^1 \&\quad + \binom{3}{2}(2t)^1(3)^2 + \binom{3}{3}(2t)^0(3)^3.\end{align*}\]

Step 4: Compute Each Term

Calculate each term independently:- \(\binom{3}{0}(2t)^3(3)^0 = 1(2t)^3 = 8t^3\).- \(\binom{3}{1}(2t)^2(3)^1 = 3 \times 4t^2 \times 3 = 36t^2\).- \(\binom{3}{2}(2t)^1(3)^2 = 3 \times 2t \times 9 = 54t\).- \(\binom{3}{3}(2t)^0(3)^3 = 1 \times 27 = 27\).

Step 5: Combine and Simplify

Combine all the terms together: \[8t^3 + 36t^2 + 54t + 27.\] This is the expansion of \((2t+3)^3\).

Key Concepts

Polynomial Expansion

Polynomial expansion is a process of transforming an expression raised to a power into a sum of terms. It makes complex expressions simpler and easier to use. Consider an expression like \((2t + 3)^3\). You can expand it using the Binomial Theorem, which breaks it down into manageable parts.

The Binomial Theorem tells us how to expand the expression \((a + b)^n\). It forms a series including powers of \(a\) and \(b\). The expanded form is a summation series:
\[ \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
The result is a polynomial with multiple terms. Each term is a component of the expansion.

When expanding polynomials, it's crucial to take note of:

• The number of terms in the expansion corresponds to \(n+1\), where \(n\) is the power in the expression \((a + b)^n\).
• The exponents of \(a\) decrease from \(n\) to 0, while the exponents of \(b\) increase from 0 to \(n\).

Understanding polynomial expansion helps in simplifying complex algebraic expressions, making them more accessible for further mathematical operations.

Binomial Coefficients

Binomial coefficients are integral elements of the binomial expansion process. They determine the multiplier for each term in a binomial expansion. Mathematically, binomial coefficients are represented as:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
`n!` (n factorial) means multiplying all positive integers up to \(n\). These coefficients play a vital role in distributing the powers of each term in the expansion correctly.

For instance, when expanding \((2t+3)^3\), we calculate the binomial coefficients for \(n = 3\) as follows:

• \(\binom{3}{0} = 1\)
• \(\binom{3}{1} = 3\)
• \(\binom{3}{2} = 3\)
• \(\binom{3}{3} = 1\)

These coefficients modify each term by indicating how many permutations of each kind of term are possible in combination.

Within polynomial expansion, binomial coefficients ensure the correct weight and structure of each resulting term, maintaining the integrity of the original binomial expression.

Algebraic Simplification

Algebraic simplification is the art of making an expression easier to work with, by reducing it to its simplest form. After expanding a binomial expression like \((2t + 3)^3\), you'll have multiple terms that need simplification.

The simplification process involves several steps:

• First, calculate the power of each term in the formula
  (for example, \((2t)^3\) becomes \(8t^3\)).
• Next, multiply the coefficients obtained from the binomial formula by these calculated terms, ensuring precise and accurate results.

This brings all terms together:
\[ 8t^3 + 36t^2 + 54t + 27 \]
The ultimate goal of algebraic simplification is to express the polynomial in the clearest, most concise way. A well-simplified expression reveals new insights and solutions, making it a vital skill in algebra and other areas of mathematics.