Problem 41 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \(y^3 + 6y^2 + 12y + 8\).

Step 1: Identify the Special Product Formula

The expression \((y + 2)^3\) is a cube of a binomial. For a binomial in the form \((a + b)^3\), the Special Product Formula is given by:\[(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\] Here, \(a = y\) and \(b = 2\).

Step 2: Apply the Formula

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

Step 3: Compute Individual Terms

Compute each term:* The first term: \(y^3\)* The second term: \(3 \times (y)^2 \times 2 = 6y^2\)* The third term: \(3 \times y \times 2^2 = 12y\)* The fourth term: \(2^3 = 8\)

Step 4: Write the Expression as a Sum of Terms

Combine all computed terms to write the expression as a sum:\[y^3 + 6y^2 + 12y + 8\]

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental concept in algebra that explains how to expand expressions of the form \((a + b)^n\). It provides a way to express each term of the expansion in terms of combinations. Understanding this theorem is crucial as it simplifies the process of expanding binomials without multiplying the binomial multiple times manually.

• Formula: The general formula is \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) represents the binomial coefficient, also known as "n choose k", which calculates how many ways \(k\) items can be chosen from \(n\) items.
• Application: This theorem is quite powerful as it applies not just to small powers like cubes, but to any positive integer \(n\). Therefore, it's very useful in higher mathematics, probability, and even in solving algebraic equations more efficiently.

Although in our scenario we are dealing specifically with the cube of a binomial, the binomial theorem provides confidence and structure for expanding many types of expressions.

Cubes of Binomials

Cubes of binomials are a special case of the binomial theorem, focusing specifically on expressions raised to the third power. This particular scenario is highly practical as cubes often appear in real-world problems like volume calculations. For instance, if you have something like \((a + b)^3\), it expands using a simplified version of the binomial formula.

• Special Product Formula: For cubing binomials, the formula becomes \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). Understanding this formula allows you to directly substitute your variables \(a\) and \(b\) — like \(y\) and \(2\) in our example — to quickly find the expanded form.
• Simplification: By calculating each term separately, as shown in our original exercise, you can easily piece together your final expression: \(y^3 + 6y^2 + 12y + 8\).

Being comfortable with cubes of binomials not only helps in expanding such expressions but also strengthens general algebra skills, making further concepts more accessible.

Algebraic Expressions

Algebraic expressions consist of variables and constants combined using arithmetic operations. They form the foundation of algebra and serve as a universal language to describe various mathematical ideas and relationships.

• Components: Algebraic expressions often include numbers, variables like \(x\) or \(y\), and operations such as addition, subtraction, multiplication, and division.
• Simplification: Simplifying algebraic expressions means reducing them to the most condensed form without changing their value. This involves combining like terms and applying algebraic identities or special product formulas.
• Advantages: Mastery of manipulating these expressions leads to solving equations, understanding patterns, and modeling real-world scenarios. The exercise we discussed is an example where understanding and applying the cube of a binomial simplified the expression quickly.

Remember, becoming comfortable with algebraic expressions is a stepping stone to more advanced topics in mathematics, hence grasping these basics is essential.