Problem 44 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(8y^3 + 36y^2 + 54y + 27\).

Step 1: Identify the Special Product Formula

The expression \((3 + 2y)^3\) can be expanded using the binomial theorem or the Binomial Expansion for cubes: \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). Here, \(a = 3\) and \(b = 2y\).

Step 2: Calculate Each Term Separately

Using the formula:- Calculate \(a^3\): \(3^3 = 27\).- Calculate \(3a^2b\): \(3(3^2)(2y) = 3(9)(2y) = 54y\).- Calculate \(3ab^2\): \(3(3)(2y)^2 = 3(3)(4y^2) = 36y^2\).- Calculate \(b^3\): \((2y)^3 = 8y^3\).

Step 3: Combine All Terms

Combine all the calculated terms to get the expanded expression: \[ 27 + 54y + 36y^2 + 8y^3 \].

Step 4: Simplify the Expression

Since there are no like terms, the expression is already simplified. Order the terms from highest to lowest power of \(y\): \[ 8y^3 + 36y^2 + 54y + 27 \].

Key Concepts

Special Product Formulas

Special product formulas are valuable shortcuts for quickly solving algebraic expressions. Instead of expanding expressions like \((a + b)^n\) by multiplying numerous times, these formulas allow you to apply already proven patterns.
In our case, the expression \((3 + 2y)^3\) can be expanded without direct multiplication by using the cube of a binomial formula:

• \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)

Here, you identify \(a = 3\)and \(b = 2y\), plug them into the formula, then evaluate separate terms one by one.
You obtain: \(a^3 = 27\), \(3a^2b = 54y\),\(3ab^2 = 36y^2\),and \(b^3 = 8y^3\).By using special product formulas, problem-solving becomes more efficient and less error-prone.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They can represent anything from simple arithmetic to complex polynomial equations.
In our example, \((3 + 2y)^3\), we're dealing with an expression that consists of both constants (3) and a term with a variable \(2y\). When you expand them using special product formulas, you re-order the terms by their degree, which simplifies understanding.
This organization makes equations easier to interpret and solve as each term corresponds to specific parts of the formula used.
Awareness in handling algebraic terms, especially those with various exponents, can greatly help in simplifying complex problems.

Polynomial Expansion

Polynomial expansion refers to the process of spreading terms of an expression that are raised to a power.
To expand a binomial like \((3 + 2y)^3\), you use the binomial theorem, which helps you write it as a simple polynomial.
In expansion, each term represents different powers of the variable and coefficients. They follow a specific pattern, based on the formula derived from the binomial theorem:

• \(a^3 = 27\)
• \(3a^2b = 54y\)
• \(3ab^2 = 36y^2\)
• \(b^3 = 8y^3\)

The polynomial generated is then \(8y^3 + 36y^2 + 54y + 27\).
This highlights the importance of understanding exponent rules and coefficient multiplication. Simplifying polynomial expressions correctly is essential, especially in higher level mathematics.