Problem 70 - Algebra A Combined Function

Question

Factor. $$ x^{3}-y^{6} $$

Step-by-Step Solution

Answer

The expression factors to \((x - y^2)(x^2 + xy^2 + y^4)\).

Step 1: Recognize the Expression as a Difference of Cubes

The expression we are dealing with, \( x^3 - y^6 \), resembles the difference of cubes. Notice that \( y^6 = (y^2)^3 \), which means \( y^6 \) can be written as a perfect cube. Therefore, the expression can be identified as \( a^3 - b^3 \) where \( a = x \) and \( b = y^2 \).

Step 2: Apply the Difference of Cubes Formula

Recall the factoring formula for a difference of cubes: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Substituting \( a = x \) and \( b = y^2 \) into this formula gives us: \( (x - y^2)(x^2 + xy^2 + (y^2)^2) \).

Step 3: Simplify the Expression

Further simplify the second factor: \( x^2 + xy^2 + (y^2)^2 \). This becomes \( x^2 + xy^2 + y^4 \), since \( (y^2)^2 = y^4 \).

Step 4: Present the Final Factored Form

The expression can now be presented in its fully factored form as \( (x - y^2)(x^2 + xy^2 + y^4) \).

Key Concepts

Difference of Cubes

The difference of cubes is a specific algebraic identity used to factor expressions of the form \(a^3 - b^3\). This identity is incredibly useful when dealing with cubic expressions, as it allows us to break them down into simpler components that are easier to work with. Here's the difference of cubes formula:

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

In the given problem, the expression \(x^3 - y^6\) can be rewritten to match this form. First, we recognize that \(y^6\) is a perfect cube because \(y^6 = (y^2)^3\). This realization lets us equate \(a = x\) and \(b = y^2\) in our formula. Substituting these values into the difference of cubes formula gives us a factored expression: \((x - y^2)(x^2 + xy^2 + y^4)\). This step is crucial as it simplifies the problem and makes the expression easier to manage and solve.

Factoring Expressions

Factoring is the process of breaking down complex expressions into simpler components, or factors, that when multiplied together give the original expression. In algebra, recognizing patterns like the difference of cubes enables students to factor expressions more efficiently.

• Factoring can involve applying known identities such as the difference or sum of cubes, difference of squares, or other polynomial identities.
• For our specific problem, factoring \(x^3 - y^6\) becomes manageable by identifying \(y^6\) as \((y^2)^3\), thus making it fit the difference of cubes form.

Breaking the expression down simplifies it into two smaller polynomial components: \((x - y^2)\) and \((x^2 + xy^2 + y^4)\). Each of these parts can be handled more easily in algebraic operations, making factoring a powerful tool in simplifying complex algebraic expressions for further calculations or problem-solving.

Polynomial Identities

Polynomial identities are important algebraic tools that provide established forms and patterns to simplify expressions. Understanding these identities allows students to manipulate and factor complex polynomial expressions with ease.The difference of cubes is just one of many polynomial identities. Others include:

• Sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
• Difference of squares: \(a^2 - b^2 = (a - b)(a + b)\)
• Perfect square trinomials and more.

By recognizing these patterns in polynomials, you can swiftly factor and simplify expressions, like converting \(x^3 - y^6\) into its factored form using the difference of cubes formula. These identities simplify algebraic calculations and are integral to solving higher-level algebra problems efficiently.