Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

(2t + 3)^3 = 8t^3 + 36t^2 + 54t + 27.

Step 1: Understand the Expression

The given problem is to simplify (2t + 3)^3. This is a binomial raised to the power of 3, which we will expand using the binomial theorem or by direct expansion.

Step 2: Apply the Binomial Theorem

The binomial theorem expresses (a + b)^n as a sum involving terms of the form C(n, k) * a^(n-k)b^k. Here, a = 2t, b = 3, and n = 3. The formula becomes (2t + 3)^3 = C(3,0)* (2t)^3 * 3^0 + C(3,1) * (2t)^2 * 3^1 + C(3,2) * (2t)^1 * 3^2 + C(3,3) * (2t)^0 * 3^3.

Step 3: Calculate Binomial Coefficients

Use the formula for binomial coefficients C(n, k) = n! / (k!(n-k)!) to find C(3,0) = 1, C(3,1) = 3, C(3,2) = 3, and C(3,3) = 1.

Step 4: Expand Each Term

Calculate each term separately: - C(3,0) * (2t)^3 * 3^0 = 1 * 8t^3 * 1 = 8t^3, - C(3,1) * (2t)^2 * 3^1 = 3 * 4t^2 * 3 = 36t^2, - C(3,2) * (2t)^1 * 3^2 = 3 * 2t * 9 = 54t, - C(3,3) * (2t)^0 * 3^3 = 1 * 1 * 27 = 27.

Step 5: Combine Terms

Add all the calculated terms together to get the final expression: 8t^3 + 36t^2 + 54t + 27.

Key Concepts

Binomial Expansion

The binomial expansion is a method used to expand expressions that are raised to a power. In our exercise, we have (2t + 3)^3, which is an example of such an expression. The binomial expansion theorem allows us to express this in terms of a sum of simpler terms.

The theorem states that \((a + b)^n\) can be expanded to:

• \(C(n, 0) \,a^n\)
• \(C(n, 1) \,a^{n-1}b\)
• \(C(n, 2) \,a^{n-2}b^2\)
• ... up to \(C(n, n) \,b^n\)

Here, \(C(n, k)\) are binomial coefficients, which we calculate for each term in the expression. This method computes each term in the expansion to form the full polynomial.
Understanding the binomial expansion is crucial in easily distributing a polynomial expression efficiently.

Polynomial Simplification

Polynomial simplification is the process of making a polynomial as simple as possible. In our exercise, after expanding \((2t + 3)^3\), we need to simplify or combine like terms to make it more concise.

To simplify the polynomial:

• Identify like terms, which are terms with the same variable raised to the same power.
• Combine these terms by performing basic arithmetic operations such as addition or subtraction.

In the expression \(8t^3 + 36t^2 + 54t + 27\), there are no like terms to combine, so it's already simplified.
Simplification helps in understanding a polynomial's form and in performing further calculations.

Mathematical Operations

Basic mathematical operations are crucial for polynomial expansion and simplification. They include addition, subtraction, multiplication, and division. In the context of our exercise, these operations help in calculating and combining the coefficients and terms from the expansion.

Here's how these operations apply:

• Addition: Used when combining like terms to simplify the polynomial.
• Multiplication: Essential when applying the binomial theorem, multiplying the coefficients \(C(n, k)\), powers of the variables, and constants together.
• Subtraction and Division: Not needed in this specific problem but often used in wider contexts for similar problems.

By mastering these operations, the expansion and simplification of polynomials become much clearer and easier to perform.