Problem 71 - Algebra and Trigonometry

Question

Factor the expression completely. $$ x^{4} y^{3}-x^{2} y^{5} $$

Step-by-Step Solution

Answer

The completely factored expression is \( x^2 y^3 (x+y)(x-y) \).

Step 1: Identify the Common Factors

To begin factoring the expression completely, observe the common elements in both terms of the expression. In both terms, the variables involved are \(x\) and \(y\). The smallest power of \(x\) across the terms is \(x^2\) and for \(y\) is \(y^3\). Thus, the greatest common factor (GCF) we can factor out is \(x^2 y^3\).

Step 2: Factor Out the Greatest Common Factor

Now, factor \(x^2 y^3\) out of the original expression: \[ x^4 y^3 - x^2 y^5 = x^2 y^3 (x^2) - x^2 y^3 (y^2)\]This simplifies the expression inside the parentheses.

Step 3: Simplify the Expression Inside the Parentheses

Next, simplify the expression inside the parentheses:\[ x^2 y^3 (x^2 - y^2) \]Now, observe the remaining expression \(x^2 - y^2\). It is a difference of squares.

Step 4: Apply the Difference of Squares

The expression \(x^2 - y^2\) can be factored further as a difference of squares:\[ x^2 - y^2 = (x+y)(x-y) \]So we rewrite the expression as:\[ x^2 y^3 (x+y)(x-y) \]

Step 5: Combine All Parts

Bring together all the parts to write the final factored form of the expression:\[ x^2 y^3 (x+y)(x-y) \] This is the completely factored form of the given expression.

Key Concepts

Greatest Common Factor

When working with polynomials, the first step in factorization is often identifying the greatest common factor (GCF). This is the largest expression that can evenly divide each term in the polynomial. Identifying it simplifies the polynomial and makes further factorization easier.

• In the expression \(x^4 y^3 - x^2 y^5\), observe each term. The term \(x^4 y^3\) consists of the variables \(x\) and \(y\) raised to powers \(4\) and \(3\) respectively. Meanwhile, \(x^2 y^5\) includes \(x\) raised to power \(2\) and \(y\) to power \(5\).
• The smallest power of \(x\) present in both terms is \(x^2\), and the smallest power of \(y\) is \(y^3\).
• Thus, the GCF is \(x^2 y^3\), which we can factor out from both terms.

By factoring out \(x^2 y^3\), you simplify the expression considerably, leading to a new expression \(x^2 y^3(x^2 - y^2)\). Recognizing the GCF is key to making polynomials easier to work with.

Difference of Squares

The concept of the difference of squares is crucial in polynomial factorization. This refers to expressions of the form \(a^2 - b^2\), which can be rewritten using the identity:
\[a^2 - b^2 = (a + b)(a - b)\]
In the given problem, after factoring out the GCF, you end up with \(x^2 - y^2\) inside the parentheses. This perfectly matches the structure of a difference of squares:

• The expression \(x^2 - y^2\) can be viewed as \(a^2 - b^2\) where \(a = x\) and \(b = y\).
• Thus, it can be factored further into two binomials \((x + y)(x - y)\).

Using the difference of squares identity simplifies factorization because it breaks down a quadratic expression into linear factors, making it much simpler to analyze or further use in computations. Recognizing this pattern quickly dovetails into efficient factorization strategies.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves reducing an expression to its most compact form. When working with complex polynomials, start by identifying common factors and special factoring patterns. Here’s how to approach simplification effectively:

• After identifying and factoring out the GCF from \(x^4 y^3 - x^2 y^5\), you're left with \(x^2 - y^2\).
• Recognize this as a difference of squares, and apply the appropriate identity: \(x^2 - y^2 = (x + y)(x - y)\).

Bringing it all together, the expression \(x^4 y^3 - x^2 y^5\) simplifies to \(x^2 y^3(x+y)(x-y)\).

Simplifying isn't just about reducing size but also preparing the expression for easier manipulation or solving, key in solving equations or evaluating functions.