Problem 56 - Algebra for College Students

Question

Use the sum-of-two-cubes or the difference-of-two-cubes pattern to factor each of the following. $$x^{6}+y^{6}$$

Step-by-Step Solution

Answer

The expression \(x^6 + y^6\) is factored as \((x^2 + y^2)(x^4 - x^2y^2 + y^4)\).

Step 1: Recognize the Expression Form

The expression, \(x^6 + y^6\), can be recognized as a sum of two cubes by first rewriting it as \((x^2)^3 + (y^2)^3\). This is because raising a squared term to the power of three generates a sixth power: \((x^2)^3 = x^{6}\) and \((y^2)^3 = y^{6}\).

Step 2: Apply the Sum of Cubes Formula

Recall the sum-of-cubes factorization formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Substitute \(a = x^2\) and \(b = y^2\) into this formula, yielding: \((x^2 + y^2)((x^2)^2 - (x^2)(y^2) + (y^2)^2)\).

Step 3: Simplify the Expression

Simplify the expression by calculating the terms inside the polynomial. For the quadratic term, calculate \((x^2)^2 = x^4\), \((x^2)(y^2) = x^2y^2\), and \((y^2)^2 = y^4\). Thus, the full factorization of the expression is: \( (x^2 + y^2)(x^4 - x^2y^2 + y^4) \).

Step 4: Verify the Factorization

To confirm the correctness of this factorization, multiply the two factors back together. When you expand \((x^2 + y^2)(x^4 - x^2y^2 + y^4)\), you should get \(x^6 + y^6\) as a result, confirming the factorization is correct.

Key Concepts

Sum of Cubes

The sum of cubes is a common factoring pattern that helps us split certain polynomial expressions into simpler factors. It applies to expressions of the form \(a^3 + b^3\). In this formula, the expression can be rewritten like so:

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

This transformation is crucial as it allows for the simplification and easier manipulation of what might at first appear to be a complicated polynomial. For example, in the exercise provided, the expression \(x^6 + y^6\) becomes a sum of cubes by rewriting each term: \((x^2)^3 + (y^2)^3\). Here, \(a = x^2\) and \(b = y^2\) are chosen such that each term is cubed.
Using this identity effectively breaks down the expression into factors that are often easier to handle, and this method is especially useful for students learning to manipulate higher-degree polynomials.

Difference of Cubes

Just like the sum of cubes, the difference of cubes is another powerful tool in factoring polynomials. Though not used in the example exercise, it complements the sum of cubes method and is equally important.
The difference of cubes applies to expressions in the form of \(a^3 - b^3\) and can be factored using the formula:

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

This formula allows you to break down an expression where two cubed values are subtracted from each other. This might occur in various polynomial problems where recognizing cube patterns becomes essential.
By learning both the sum and difference of cubes formulas, you can easily factor a wider range of expressions, increasing your ability to solve polynomial equations.

Polynomial Expressions

A polynomial expression is an expression that comprises variables, coefficients, and non-negative integer exponents. Understanding how to manipulate and factor these expressions is crucial for many areas in mathematics.
When working with polynomial expressions, recognizing specific patterns, such as the sum and difference of cubes, allows you to factor effectively and simplify the process of solving equations.

• Polynomials can vary greatly in complexity, ranging from simple linear expressions to higher-degree expressions like the one in the exercise \(x^6 + y^6\).
• Factoring is important because it reduces expressions into their simplest form, making them easier to work with and solve within various mathematical contexts.

With a solid understanding of polynomial expressions and various factoring techniques, you'll have a powerful toolkit for tackling complex mathematical problems. Make sure to practice these techniques regularly to master them.