Problem 62 - Algebra 2

Question

REVIEW \(27 x^{3}+y^{3}=\) \(\mathbf{A}(3 x+y)(3 x+y)(3 x+y)\) \(\mathbf{B}(3 x+y)\left(9 x^{2}-3 x y+y^{2}\right)\) \(\mathbf{C}(3 x-y)\left(9 x^{2}+3 x y+y^{2}\right)\) \(\mathbf{D}(3 x-y)\left(9 x^{2}+9 x y+y^{2}\right)\)

Step-by-Step Solution

Answer

The correct answer is option B.

Step 1: Identify the Given Expression

The expression we need to factor is \(27x^3 + y^3\). This is in the form of a sum of cubes.

Step 2: Recall Sum of Cubes Formula

The sum of cubes formula is given by \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). We need to rewrite \(27x^3\) and \(y^3\) in terms of cubes.

Step 3: Express as Cubes

Recognize \(27x^3 = (3x)^3\) and \(y^3 = y^3\). Thus, the expression is \((3x)^3 + y^3\). Here, \(a = 3x\) and \(b = y\).

Step 4: Apply Sum of Cubes Formula

By substituting into the sum of cubes formula, we get \((3x + y)((3x)^2 - 3x \, y + y^2)\).

Step 5: Simplify the Second Term

Simplify \((3x)^2 - 3x \, y + y^2\) to get \(9x^2 - 3xy + y^2\).

Step 6: Match with Given Choices

The factored form \((3x + y)(9x^2 - 3xy + y^2)\) matches choice \(\mathbf{B}\).

Key Concepts

Sum of Cubes

The expression provided, \(27x^3 + y^3\), is an example of the "sum of cubes." This mathematical concept involves expressions of the form \(a^3 + b^3\), which can be factored using a specific formula. The formula for the sum of cubes is:

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

To factor using this formula, it is important to express each term as a perfect cube. In our example, \(27x^3\) is the cube of \(3x\), and \(y^3\) is already a perfect cube with \(b = y\). With these values, we can directly apply the sum of cubes formula, substituting \(a = 3x\) and \(b = y\).
Following this substitution provides a straightforward way to fact the expression neatly into \((3x + y)(9x^2 - 3xy + y^2)\).
This approach simplifies potentially complex expressions into cleaner, more manageable factors.

Factoring Techniques

Factoring techniques are essential tools in algebra to simplify expressions and solve equations more efficiently. For expressions like the sum of cubes, we specifically use the sum of cubes formula to break down complex polynomials into simpler terms. Here are a few key steps when using this technique:

• Identify the expression as a sum of cubes.
• Express each term in its cubic form.
• Apply the sum of cubes formula.

In our case, recognizing that \(27x^3 + y^3\) is written as \((3x)^3 + y^3\) allows us to set up for factoring by substituting \(a = 3x\) and \(b = y\) into the formula.
This technique not only aids in simplifying expressions but also plays a critical role in later algebraic problem-solving tasks, such as finding roots of polynomial equations.
Mastering such techniques makes handling algebraic expressions far less intimidating.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition, multiplication, etc.) that collectively represent a value. In the polynomial world, expressions like \(27x^3 + y^3\) serve as prime examples of algebraic expressions requiring special methods for simplification.
Polynomials are one type of algebraic expression that often benefit from advanced factoring techniques, like the sum of cubes, to reduce their complexity.
Understanding algebraic expressions involves recognizing their components:

• Coefficients: These are the numerical part, like 27 in \(27x^3\).
• Variables: Symbols that represent numbers, such as \(x\) and \(y\).
• Exponents: These indicate the power to which the variable is raised, as in \(x^3\).

Manipulating these elements through techniques like factoring simplifies problem-solving and allows for clearer insight into the behavior of the expressions. Thus, a firm grasp of algebraic expressions and their manipulation is crucial in math studies.