Problem 55 - Elementary Algebra

Question

Factor completely. $$ 54 x 4 y-2 x y 4 $$

Step-by-Step Solution

Answer

The expression factors to \( 2xy(3x - y)(9x^2 + 3xy + y^2) \).

Step 1: Simplify the Expression

First, let's rewrite the given expression stepping through any confusion due to formatting and symbols. We have:\[ 54x^4y - 2xy^4 \]

Step 2: Factor Out the Greatest Common Factor (GCF)

Identify the greatest common factor for the terms in the expression. Here, the GCF is \( 2xy \). Factor \( 2xy \) out of each term:\[ 54x^4y = 2xy imes 27x^3 \]\[ 2xy^4 = 2xy imes y^3 \]Thus, we can factor out the GCF:\[ 2xy(27x^3 - y^3) \]

Step 3: Recognize the Difference of Cubes

The expression inside the parentheses \( 27x^3 - y^3 \) is a difference of cubes. This allows further factoring using the difference of cubes formula:\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]Here, let \( a = 3x \) and \( b = y \), then apply the formula.

Step 4: Apply the Difference of Cubes Formula

Substitute \( a = 3x \) and \( b = y \) into the difference of cubes formula and factor:\[ 27x^3 - y^3 = (3x - y)((3x)^2 + 3x imes y + y^2) \]Compute the expression:\[ = (3x - y)(9x^2 + 3xy + y^2) \]

Step 5: Final Answer: Combine All Factors

Combine all parts to present the fully factored expression:\[ 54x^4y - 2xy^4 = 2xy(3x - y)(9x^2 + 3xy + y^2) \]

Key Concepts

Greatest Common Factor

When factoring polynomials, the greatest common factor (GCF) is the largest term that can be evenly divided from all terms of the polynomial. Identifying the GCF is the first step in the factoring process and can simplify the expression.

To find the GCF, look for numerical coefficients and variables common to each term of the polynomial. In the example given, we have two terms:

• 54x^4y
• 2xy^4

The GCF here is found by identifying the smallest power of each common factor. For numbers, it's the greatest number that divides both 54 and 2, which is 2. For variables, such as x and y, it's the one with the lowest exponent shared across the terms, so x and y both to the power of 1. Hence, the GCF of the expression is 2xy.

By factoring out the GCF, the expression becomes simpler, allowing easier manipulation or further factoring, as shown in subsequent steps.

Difference of Cubes

The difference of cubes formula is a handy tool in algebra that allows us to factor specific types of polynomial expressions. It is applicable when you encounter the subtraction of two perfect cubes.

The formula is:\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

Recognizing when this formula can be used is crucial. In our example, after factoring out the GCF, we are left with \( 27x^3 - y^3 \), which matches the structure of a^3 - b^3.

• Here, 27x^3 is the cube of 3x (since \((3x)^3 = 27x^3\)).
• Similarly, y^3 is the cube of y (as \(y^3 = y^3\)).

By letting \(a = 3x\) and \(b = y\), the formula can be directly applied.

This breaks down the expression into a binomial \((3x - y)\) and a trinomial \((9x^2 + 3xy + y^2)\), completing the factorization process.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (such as +, –, *, /). These expressions are the foundation of algebra and are used to represent real-world problems and mathematical relationships.

Algebraic expressions can be manipulated through various operations, including simplifying, expanding, and factoring. Factoring, like in our example, involves breaking down expressions into products of simpler factors. Understanding how to manipulate these expressions is essential in solving equations, graphing functions, and more.

When dealing with algebraic expressions:

• Identify the terms and coefficients.
• Use factoring techniques like the GCF or special formulas such as the difference of cubes.
• Reassemble the expression to its most simplified or desired form.

Mastering these basic operations on algebraic expressions saves time and enhances problem-solving skills in advanced mathematics. Remember, practice is key to becoming comfortable with such concepts.