Problem 44 - College Algebra and Calculus: An Applied Approach

Question

Find the product.\((3 x+2 y)^{3}\)

Step-by-Step Solution

Answer

The expanded form of \((3x+2y)^3\) is \(27x^3 + 54x^2y + 36xy^2 + 8y^3\).

Step 1: Write down the binomial cube formula

This is a problem of expanding a cube of a binomial. The formula is \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)

Step 2 : Substitute the values of the binomial formula

Substitute \(a = 3x\) and \(b = 2y\) into the formula. This gives: \((3x+2y)^3 = (3x)^3 + 3(3x)^2*(2y) + 3(3x)*(2y)^2 + (2y)^3\).

Step 3: Simplify

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

Key Concepts

Polynomial Expansion

Expanding polynomials is an essential skill in algebra that involves breaking down expressions raised to a power into a sum of terms. Through expansion, we transform a compact expression into one that is more spread out but reveals the individual components. This can be achieved using formulas such as the binomial theorem for binomials.

• Single Variable Example: If you have a simple expression like \( (x+1)^2 \), expanding it means writing it as \( x^2 + 2x + 1 \).
• Multi-variable Example: Polynomial expansion with variables can look like \( (a + b + c)^3 \), which gets transformed into multiple terms like \( a^3 + b^3 + c^3 + ... \).

Understanding polynomial expansion helps in simplifying complex expressions and is vital in solving equations effectively. In particular, when dealing with powers greater than two, like cubes, expansion often uses specific formulas to break down quickly.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. They can range from simple to very complex expressions, pivotal in forming and solving equations.

• Variables: Symbols like \( x \) and \( y \) that can represent unknown quantities or varied values.
• Constants: The actual numbers used in expressions, e.g., 2, 3, 5.
• Operations: Basic math like \(+, -, *, /\) that combine the numbers and variables.

When you manipulate algebraic expressions, like using substitution or simplification, you control and change these components to achieve another expression or to solve an equation. For example, in the original exercise, observing how each term in \((3x)^3 + 3(3x)^2(2y) + 3(3x)(2y)^2 + (2y)^3\) is derived requires understanding how to manage these elements properly to ensure the simplification is correct.

Binomial Cube Formula

The binomial cube formula is a special algebraic identity that simplifies the expansion of a binomial raised to the third power. Here's how you can use it:

• Formula: \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)
• Components: Breakdown of the formula:
  • \(a^3\) represents the cube of the first term.
  • \(3a^2b\) and \(3ab^2\) highlight the mixed terms with coefficients from expansion.
  • \(b^3\) signifies the cube of the second term.

To apply this, like in the problem \((3x+2y)^3\), you substitute \(a = 3x\) and \(b = 2y\) into the formula.
Then, each term is computed separately and combined to arrive at the expanded form. Understanding this formula helps unravel explanatory breakdowns, particularly in high school algebra problems, where binomial expressions frequently appear. It is a powerful tool for simplifying equations involving polynomial expressions and enhancing computational fluency.