Problem 13 - Introductory Algebra for College Students

Question

Before getting to multiple-step factorizations, let's be sure that you are comfortable with exercises requiring only one of the factoring techniques. Factor each polynomial. $$27 x^{3} y^{3}+8$$

Step-by-Step Solution

Answer

So, the factored form of the polynomial is \((3xy + 2)(9x^2y^2 - 6xy + 4)\)

Step 1: Express the Polynomial as a Sum of Cubes

First, rewrite the polynomial \(27x^3y^3 + 8\) in the form \(a^3+b^3\). That gives \((3xy)^3 + 2^3\).

Step 2: Use the Formula for Factoring a Sum of Cubes

The sum of cubes can be factored using the formula \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). Substitute \(a = 3xy\) and \(b = 2\) into the formula. That gives \((3xy + 2) [(3xy)^2 - 2*3xy + 2^2]\).

Step 3: Simplify the Result

Now, simplify the expression \((3xy + 2) [(3xy)^2 - 2*3xy + 2^2]\), this yields \((3xy + 2)(9x^2y^2 - 6xy + 4)\).

Key Concepts

Sum of Cubes

Understanding the concept of sum of cubes is essential when dealing with certain types of polynomial expressions. In algebra, a sum of cubes refers to an expression of the form \( a^3 + b^3 \) where \( a \) and \( b \) are algebraic expressions. To factorize this expression, we use the identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). This is a special factorization formula that allows us to break down the sum of two cubes into the product of a binomial and a trinomial.

Applying this to our example, we have \(27x^3y^3 + 8\text{. This can be rewritten as } (3xy)^3 + 2^3\), identifying \(3xy\) as \(a\) and \(2\) as \(b\). By using the sum of cubes formula, we factor it into \( (3xy + 2)((3xy)^2 - 3xy*2 + 2^2) \), which can be further simplified.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. In the context of the given exercise, \(27x^3y^3 + 8\) is an algebraic expression with terms that are products of numerical coefficients and powers of variables.

To work with algebraic expressions efficiently, recognizing patterns and structures, such as the sum of cubes, is critical. This skill allows us to manipulate and simplify expressions by applying factorization techniques. In our example, recognizing that \(27x^3y^3\) and \(8\) are both perfect cubes \( (3xy)^3 \) and \(2^3\), respectively) is the key step that leads to applying the appropriate factoring technique.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as the product of its factors, which are usually simpler polynomials. The goal is to decompose a complicated expression into a product of polynomials of lower degree.

There are different methods of factoring polynomials, depending on their form. For example, to factor a sum of cubes, we use its unique formula. The exercise \(27x^3y^3 + 8\) showcases this, where after recognizing the structure, we apply the sum of cubes formula to factor it into a binomial and trinomial product. Polynomial factorization is not only a fundamental concept in algebra but also an invaluable tool in solving equations and simplifying expressions.

Simplifying Expressions

The process of simplifying expressions involves reducing an algebraic expression to its simplest form. This means combining like terms, reducing fractions, and factoring expressions wherever possible. Simplification can make expressions easier to understand and work with and is an important step when solving algebraic problems.

In our example, after using the sum of cubes formula to factor \(27x^3y^3 + 8\), we arrive at \(3xy + 2)(9x^2y^2 - 6xy + 4)\). This product is the simplified form of the original expression and cannot be reduced any further using basic arithmetic operations or factorization techniques. Remember, properly simplifying expressions is key to success in algebra and beyond, as it sets the stage for solving more complex mathematical problems.