Problem 14 - 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. $$216 x^{3} y^{3}+125$$

Step-by-Step Solution

Answer

The factored form of \(216x^{3}y^{3} + 125\) is \((6xy + 5)(36x^{2}y^{2} - 30xy + 25)\)

Step 1: Identify the Terms as Cubes

The given polynomial is \(216x^{3}y^{3} + 125\). We can rewrite it as \((6xy)^{3} + 5^{3}\). So, \(a = 6xy\) and \(b = 5\).

Step 2: Substitute Values into Sum of Cubes Formula

We substitute the values of \(a\) and \(b\) into the formula \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\). So, it will be \((6xy + 5)((6xy)^{2} - 6xy*5 + 5^{2})\).

Step 3: Simplify the Expression

Simplify each term in the second parentheses by applying the powers and performing the operations: \( (6xy + 5)(36x^{2}y^{2} - 30xy + 25)\).

Key Concepts

Sum of Cubes

When dealing with algebraic expressions, one common factoring task is recognizing the sum of cubes. This technique is handy when you have a polynomial made up of terms that are perfect cubes. In our EXERCISE, we have a polynomial expression:

• \(216x^{3}y^{3} + 125\)

Breaking it down, you can identify each part as a cube: \((6xy)^{3}\) and \(5^{3}\). Here, it's important to remember the sum of cubes formula, which is:

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

This formula helps break down and simplify the polynomial. By substituting \(a = 6xy\) and \(b = 5\), you can efficiently factor these types of problems.

Factoring Techniques

Greater understanding of factoring techniques can unravel complex polynomials into manageable parts. There are several methods other than the sum of cubes, such as:

• Difference of squares
• Quadratic factoring
• Grouping
• Using the greatest common factor (GCF)

In this solution scenario, the sum of cubes formula is pertinent. Recognizing it requires identifying terms as cubes and then employing the specific formula to factor them. It involves reformatting the expression to fit a formulaic pattern, simplifying computation. This is the basis for attacking more complicated factoring exercises. The formula makes it easier by breaking it into a binomial and trinomial, making further operations straightforward.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. Understanding them is fundamental for factoring tasks like in this EXERCISE. Our expression \(216x^{3}y^{3} + 125\) comprises both:

• Cubic terms like \(216x^{3}y^{3}\)
• Constant terms like \(125\)

The challenge often lies in manipulating these expressions through operations such as factoring. Recognizing patterns, like cubes, makes this easier. Algebraic expressions allow you to apply operations to variables, leading to simplified, factored forms suitable for further calculations or solving equations. The key point is to see patterns and use factoring formulas that simplify complex-looking polynomials.

Polynomial Expressions

Polynomial expressions involve sums of multiple algebraic terms. In our case, the expression \(216x^{3}y^{3} + 125\) includes just two terms. Understanding that each term can be a factorable unit means recognizing types like cubes. Polynomials are summed in powers of variables, and factoring them involves breaking them down from highest degree terms down to simpler expressions.

• Here, recognizing cubes led to using a special case formula: the sum of cubes.
• Each polynomial expression can have its factoring technique.

With polynomials, always look for potential factoring methods to simplify them, making equations easier to handle. With practice, various polynomial forms become much easier to manage, leading to efficient problem-solving.