Problem 55 - Elementary Algebra

Question

Factor completely. $$ y 3-27 $$

Step-by-Step Solution

Answer

The complete factorization is \( (y - 3)(y^2 + 3y + 9) \).

Step 1: Identify the Expression

The given expression is \( y^3 - 27 \). It is a difference of cubes.

Step 2: Recognize the Difference of Cubes Formula

The formula for factoring a difference of cubes \( a^3 - b^3 \) is \( (a - b)(a^2 + ab + b^2) \).

Step 3: Determine \( a \) and \( b \)

In the expression \( y^3 - 27 \), \( a = y \) and \( b = 3 \), since \( 27 = 3^3 \).

Step 4: Apply the Difference of Cubes Formula

Substitute \( a = y \) and \( b = 3 \) into the formula to get \( (y - 3)(y^2 + 3y + 9) \).

Step 5: Confirm the Factorization

Verify that \( (y - 3)(y^2 + 3y + 9) \) is the correct factorization by expanding it back to the original expression \( y^3 - 27 \).

Key Concepts

Difference of Cubes

The concept of the difference of cubes is a special technique used in algebra to simplify expressions like the one given in the exercise: \( y^3 - 27 \). Here, we are dealing with cubes because both terms can be expressed as perfect cubes. Specifically, \( y^3 \) is already a cube of \( y \) and \( 27 \) is a cube of \( 3 \), since \( 3^3 = 27 \).

The formula for the difference of cubes is used to factor expressions of the form \( a^3 - b^3 \). It states that:

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

When using this formula, it is essential to identify the two values, \( a \) and \( b \), that when cubed give us the terms of our expression. In our exercise, \( a = y \) and \( b = 3 \).

Applying this formula correctly will factor the difference of cubes completely, offering a neat and simpler way to represent complex algebraic expressions.

Algebra Expressions

An algebraic expression is a combination of numbers, variables, and operations. In the exercise presented \( y^3 - 27 \), the expression consists of a variable term \( y^3 \) and a constant \( 27 \).

Understanding algebraic expressions is crucial in performing operations like factoring. Recognizing the specific structure of these expressions, such as cubes or other polynomial forms, allows for the appropriate application of algebraic rules.

When manipulating algebraic expressions, one often needs to identify patterns or formulas that simplify the expression. Here, realizing that \( y^3 - 27 \) is a difference of cubes is the key to choosing the correct factoring method. The variable term and the constant can both be reduced to simpler forms by using known algebraic identities, making operations like factorization possible.

Polynomial Factorization

Polynomial factorization refers to breaking down a polynomial into simpler components (factors) that, when multiplied together, give back the original polynomial. The focus of this exercise is on the factorization of a particular type of polynomial known as the difference of cubes.

In general, to factor a polynomial effectively, one must:

• Recognize the structure or form of the polynomial.
• Identify possible factoring formulas or methods.
• Apply the appropriate formula to rewrite the polynomial as a product of factors.

For the expression \( y^3 - 27 \), the difference of cubes formula is used. By applying this formula, we transform the expression into \((y - 3)(y^2 + 3y + 9)\). Unlike quadratic or linear polynomials, cubes require their own special approach based on the distinct powers involved.

This example demonstrates how polynomial factorization helps simplify expressions and solve equations effectively by breaking them into their most basic elements.