Problem 70 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \((3+2 y)^{3}\)

Step-by-Step Solution

Answer

\(8y^3 + 36y^2 + 54y + 27\)

Step 1: Identify the Special Product Formula

Recognize that the expression \((3 + 2y)^3\) is a cubic binomial expression. The special product formula for the cube of a binomial \((a+b)^3\) is: \(a^3 + 3a^2b + 3ab^2 + b^3\). Here, \(a = 3\) and \(b = 2y\).

Step 2: Apply the Special Product Formula

Substitute \(a = 3\) and \(b = 2y\) into the formula: \((3 + 2y)^3 = 3^3 + 3 \times 3^2 \times 2y + 3 \times 3 \times (2y)^2 + (2y)^3\).

Step 3: Calculate Each Term

Compute each part of the expanded expression: - First term: \(3^3 = 27\). - Second term: \(3 \times 9 \times 2y = 54y\). - Third term: \(3 \times 3 \times 4y^2 = 36y^2\). - Fourth term: \((2y)^3 = 8y^3\).

Step 4: Combine the Terms

Combine the calculated terms to get the expanded expression: \(27 + 54y + 36y^2 + 8y^3\). Reorder the terms in descending order: \(8y^3 + 36y^2 + 54y + 27\).

Key Concepts

Cubic Binomial

To understand what a cubic binomial is, let's first break down the term into its parts. A *binomial* is an algebraic expression that consists of exactly two terms, which are usually separated by a plus or minus sign. Examples of simple binomials include \((x + 2)\) or \((3y - 5)\).
A *cubic binomial*, then, is simply a binomial that has been raised to the power of three. This means that instead of just being a sum or difference, the entire expression is cubed. A common example is \((a + b)^3\), where both \(a\) and \(b\) are parts of the binomial, and the cube denotes that we multiply the entire binomial by itself three times.
Cubing a binomial is special because it expands into more terms than the original expression. Using the special product formula that handles cubic binomials makes it easier to carry out the computation without multiplying the entire expression manually. This approach not only reduces the complexity of the problem but also ensures accuracy while working with polynomial expansions.

Algebraic Expressions

Algebraic expressions are the foundation of algebra and represent mathematical phrases involving numbers, variables, and operation symbols. Expressions form the building blocks for equations and other more detailed analyses in mathematics. For example, an algebraic expression may look like \((3 + 2y)^3\), which we'll explore using specific methods through this guide.
Some key components of algebraic expressions include:

• *Variables*: Symbols, such as \(x\) or \(y\), that can represent unspecified numbers. In terms of your exercise, \(y\) plays this part.
• *Constants*: Numbers like \(3\) in the expression \((3 + 2y)^3\).
• *Operation symbols*: Signs like \(+\), \(-\), \(\times\), and \(\div\), which indicate the operations being performed within the expression.

Dealing with algebraic expressions requires understanding these components and knowing how to manipulate them properly. For instance, simplifying \((3 + 2y)^3\) involves recognizing that it's a cubic binomial, applying a special product formula, and simplifying the resulting expression. This derails potential errors in mathematical operations and offers a streamlined path to obtain the final result more efficiently than manual multiplication.

Polynomial Expansion

Polynomial expansion is an essential concept when working with algebraic expressions, and it specifically involves rewriting expressions in expanded form. This transformation is crucial for simplifying complex expressions and facilitating easier computation. A polynomial is a mathematical expression consisting of multiple terms, like \((3 + 2y)^3\) when expanded.
Here's how polynomial expansion works in practice using our special product formula: When you expand a cubic binomial, you apply the formula \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). This is an efficient way to transform the original expression into a polynomial that can be easily manipulated and evaluated.
For example:

• The first term \(3^3\) within \((3 + 2y)^3\) expands to \(27\).
• The subsequent terms follow the formula, incorporating the constants \(a = 3\) and \(b = 2y\) to give us all terms necessary for the expansion.
• When all terms are brought together and ordered, you have a neatly expanded polynomial \(8y^3 + 36y^2 + 54y + 27\).

This expansion not only provides clarity but also brings out the structure of the expression, aiding in further mathematical analysis or application.