Problem 67 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\(y^3 + 6y^2 + 12y + 8\)

Step 1: Recognize the Special Product Formula

The expression \((y+2)^3\) can be expanded using the binomial theorem formula known as the cube of a binomial: \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). Here, \(a = y\) and \(b = 2\).

Step 2: Apply the Cube of a Binomial Formula

Substitute \(a = y\) and \(b = 2\) into the formula: \(a^3 + 3a^2b + 3ab^2 + b^3\).- Calculate \(a^3 = y^3\).- Calculate \(3a^2b = 3(y^2)(2) = 6y^2\).- Calculate \(3ab^2 = 3(y)(2)^2 = 12y\).- Calculate \(b^3 = 2^3 = 8\).

Step 3: Simplify the Expression

Combine all terms: \(y^3 + 6y^2 + 12y + 8\). There are no like terms, so this is the simplified expression.

Key Concepts

Cube of a Binomial

The cube of a binomial involves expanding an expression in the form of \((a + b)^3\), where you apply the binomial theorem. This is a powerful and useful tool for expanding polynomial expressions. The formula for the cube of a binomial is:\[(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\]In this formula:

• \(a^3\) is the cube of the first term.
• \(3a^2b\) represents three times the square of the first term multiplied by the second term.
• \(3ab^2\) signifies three times the product of the first term and the square of the second term.
• \(b^3\) is the cube of the second term.

Each part of the formula builds on understanding how polynomials expand. Once you identify the pattern, you can quickly apply it to any expressions following the same structure. Regardless of the variables or numbers, this rule stays consistent, helping in simplifying complex expressions.

Special Product Formula

The Special Product Formula is a mathematical identity used to simplify the expansion of certain expressions, such as binomials. These formulas allow you to skip the tedious process of distributing and providing a quick path to an expanded form. For example, the cube of a binomial is one type of Special Product Formula. Learning these formulas can save time and help with accuracy in solving algebraic problems:

• Squares of a Binomial: \((a+b)^2 = a^2 + 2ab + b^2\)
• Difference of Squares: \(a^2 - b^2 = (a+b)(a-b)\)
• Cube of a Binomial: \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)

These formulas make the process of working with polynomials much smoother. Memorizing them not only aids in exams but also increases your fluency in algebraic manipulation. They reveal the relationships between terms in an expression and help make predictions about the patterns.

Algebraic Expressions

Algebraic expressions are mathematical phrases combining numbers, variables, and operational symbols. They form the foundation for all of algebra and are used to represent real-world scenarios and complex patterns. In the context of binomials and their expansions, algebraic expressions allow for expressing complicated problems in a simpler form. For instance, in \((y+2)^3\), you can see two terms \(y\) and 2 combined in a binomial expression. Applying the Special Product Formula helps expand these expressions in a more digestible format that is easier to simplify.Breaking down complex expressions into manageable parts is key. By understanding each component:

• Terms: Elements separated by plus or minus signs.
• Coefficients: Numerical factors of terms, like 6 in 6y^2.
• Variables: Symbols like y that stand in for unknown values.
• Exponents: Show the power to which a variable is raised, e.g. \(y^3\).

You gain clarity on how expressions interact and transform. Mastery of algebraic expressions enhances your algebra skills and simplifies the study of higher-level math.