Problem 21 - Precalculus: Functions and Graphs

Question

Express as a polynomial. $$(3 x+2 y)^{2}(3 x-2 y)^{2}$$

Step-by-Step Solution

Answer

The polynomial is \(81x^4 - 72x^2y^2 + 16y^4\).

Step 1: Recognize the expression structure

Notice that the initial expression \((3x + 2y)^2(3x - 2y)^2\) can first be simplified using the identity \((a^2)(b^2) = (ab)^2\). Here, think of \(a = 3x + 2y\) and \(b = 3x - 2y\). Hence, the expression simplifies to \(((3x + 2y)(3x - 2y))^2\).

Step 2: Apply the difference of squares formula

Recall the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\). Apply this to \((3x+2y)(3x-2y)\) with \(a = 3x\) and \(b = 2y\), giving \((3x)^2 - (2y)^2\). Simplifying, we get \(9x^2 - 4y^2\).

Step 3: Expand further using the square of a binomial

Now that we have \((9x^2 - 4y^2)^2\), we need to expand it: \((a-b)^2 = a^2 - 2ab + b^2\), where \(a = 9x^2\) and \(b = 4y^2\). Calculate it as follows: \((9x^2)^2 - 2(9x^2)(4y^2) + (4y^2)^2\).

Step 4: Calculate each term

Calculate the individual terms: 1. \((9x^2)^2 = 81x^4\) 2. \(-2(9x^2)(4y^2) = -72x^2y^2\) 3. \((4y^2)^2 = 16y^4\).

Step 5: Combine the results into a polynomial

Combining the results of the calculations, the expanded polynomial is \(81x^4 - 72x^2y^2 + 16y^4\).

Key Concepts

Difference of Squares

The difference of squares is a powerful algebraic identity that simplifies expressions like

• \[(a + b)(a - b) = a^2 - b^2\]

This formula is incredibly useful for breaking down complex expressions into simpler terms.
In our original problem, we applied this formula to

• \((3x + 2y)(3x - 2y)\)

By doing so, we transformed this expression into

• \((3x)^2 - (2y)^2\)

which simplifies further to

• \[9x^2 - 4y^2\]

Recognizing and applying the difference of squares can often make polynomial work much faster.

Square of a Binomial

The square of a binomial is another fundamental algebraic identity. It lets you expand expressions like

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

In our problem, once we simplified using the difference of squares, we were left with

• \((9x^2 - 4y^2)^2\)

To expand this using the square of a binomial formula, we set

• \(a = 9x^2\)
• and \(b = 4y^2\)

This resulted in the expression

• \((9x^2)^2 - 2(9x^2)(4y^2) + (4y^2)^2\)

Breaking down the square of a binomial can significantly simplify the expansion process.

Simplification Techniques

Simplification techniques are essential for making algebraic expressions easier to handle. Let's review the process of simplification we used.
The first step was recognizing the structure of the expression

• \((3x + 2y)^2(3x - 2y)^2\)

Here, noticing that it could be rewritten using the identity

• \( (a^2)(b^2) = (ab)^2\)

led us to

• \(((3x + 2y)(3x - 2y))^2\)

This simplification paved the way for applying other techniques like the difference of squares and the square of a binomial, both of which resulted in a dramatically simplified form.
Simplification not only makes calculations more manageable but also helps in identifying further algebraic opportunities.

Expression Structure

Understanding the structure of an expression helps identify the best strategies for simplification or expansion. The expression structure tells you what kind of algebraic identities or techniques might apply. In the given problem, the expression

• \((3x + 2y)^2(3x - 2y)^2\)

was immediately an indicator for using the identity

• \((a^2)(b^2) = (ab)^2\)

Recognizing that the expression is a product of squares, we simplified it to

• \(((3x + 2y)(3x - 2y))^2\)

Each part of the expression suggests potential algebraic strategies. Sometimes it means spotting a pattern or applying a known formula. By understanding expression structures, you can approach seemingly complex algebra tasks with more confidence.

Algebraic Identities

Algebraic identities are like shortcuts for transforming and simplifying expressions. There are many identities, but some are used more frequently, like:

• The difference of squares: \[(a + b)(a - b) = a^2 - b^2\]
• The square of a binomial:\[(a + b)^2 = a^2 + 2ab + b^2\]
• \[(a - b)^2 = a^2 - 2ab + b^2\]

Recognizing these identities allows you to simplify expressions quickly. In our problem, we leveraged these identities to reduce a complex product into manageable steps, ultimately yielding the polynomial

• \[81x^4 - 72x^2y^2 + 16y^4\]

By frequently practicing these identities, you can solve algebraic problems efficiently and accurately.