Problem 34 - Elementary Algebra

Question

Factor completely. $$ -18 x 3 y 3+32 x y $$

Step-by-Step Solution

Answer

The expression factors to \(2xy(3xy - 4)(3xy + 4)\).

Step 1: Identify the Greatest Common Factor (GCF)

To begin factoring the expression \(-18x^3y^3 + 32xy\), we first identify the greatest common factor of the terms. The common factors include numbers and variables. The numerical GCF of \(-18\) and \(32\) is \(2\). For the variables, the smallest power in common for \(x\) is \(x\) and for \(y\) is \(y\). Thus, the GCF of the entire expression is \(2xy\).

Step 2: Factor Out the GCF

Next, we factor \(2xy\) out of each term in the expression:\[-18x^3y^3 + 32xy = 2xy(-9x^2y^2 + 16)\].

Step 3: Express the Inside as a Difference of Squares

Examine \(-9x^2y^2 + 16\) inside the parentheses. Notice that each term is a perfect square: \(-9x^2y^2 = -(3xy)^2\) and \(16 = (4)^2\). The expression can be rewritten as a difference of squares: \((-3xy)^2 - (4)^2\).

Step 4: Apply the Difference of Squares Formula

Use the formula \(a^2 - b^2 = (a - b)(a + b)\) for the expression \((-3xy)^2 - 4^2\): - Set \(a = 3xy\) and \(b = 4\), then apply the formula:\((3xy - 4)(3xy + 4)\).

Step 5: Substitute Back into the Factored Expression

Replace the factored form of \(-9x^2y^2 + 16\) back into the original expression:\[2xy(-9x^2y^2 + 16) = 2xy(3xy - 4)(3xy + 4)\].

Step 6: Verification

Verify by redistributing: \[2xy(3xy - 4)(3xy + 4) = 2xy[(3xy)^2 - 4^2] = 2xy(-9x^2y^2 + 16)\], confirming the factorization is correct.

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor, often abbreviated as GCF, is a key concept in algebra that refers to the largest factor that can evenly divide each term in a polynomial expression. To find the GCF in an algebraic expression, we need to consider both numerical coefficients and variables present in the terms.

When analyzing the expression \(-18x^3y^3 + 32xy\), we first look at the coefficients \(-18\) and \(32\). The largest number that can evenly divide both is \(2\). For the variables, we identify the smallest powers that appear in each term: for \(x\) it is \(x^1\), and for \(y\) it is \(y^1\).

• Identify the numerical GCF: 2
• Identify the lowest power of each repeating variable: \(x\) and \(y\)
• Combine these to form the complete GCF: \(2xy\)

Extracting the GCF simplifies the expression and makes further factorization possible.

Difference of Squares

The difference of squares is another important algebraic concept often used to simplify polynomial expressions. It involves rewriting an expression of the form \(a^2 - b^2\) as \((a - b)(a + b)\).

In the exercise, after factoring out the GCF from the expression \(-18x^3y^3 + 32xy\), you are left with \(-9x^2y^2 +16\). Notice that both \(-9x^2y^2\) and \(16\) are perfect squares:

• \(-9x^2y^2 = (-3xy)^2\)
• \(16 = 4^2\)

Rewriting this expression using the difference of squares formula gives us \((-3xy - 4)(-3xy + 4)\), allowing further simplification. Applying this technique can greatly aid in factorization of complex expressions.

Polynomial Expressions

A polynomial expression is made up of variables raised to a power (exponents), combined using addition or subtraction. Factoring polynomials is an essential technique in algebra for simplifying expressions and solving equations. It involves breaking down the polynomial into products of simpler expressions.

Consider the original polynomial expression \(-18x^3y^3 + 32xy\). This expression has two terms, indicating it's a binomial. With polynomial expressions, the goal is often to pull out common factors, then look for any special algebraic identities, such as difference of squares, that can help further simplify the expression.

• Break complex expressions into simpler pieces
• Use common identities, such as difference of squares, to aid in factorization
• Once factored, polynomial expressions can be easier to analyze or solve

This method not only simplifies the expression but also prepares it for more advanced operations or to find its roots, which are useful in solving equations.