Problem 30 - Elementary Algebra

Question

Factor completely. $$ 16 a 4-81 b 4 $$

Step-by-Step Solution

Answer

\((2a - 3b)(2a + 3b)(4a^2 + 9b^2)\) is the fully factored expression.

Step 1: Recognize the Structure

The expression given is \(16a^4 - 81b^4\). Notice that this is a difference of squares. It can be rewritten as \((4a^2)^2 - (9b^2)^2\).

Step 2: Apply Difference of Squares Formula

The difference of squares formula states that \(x^2 - y^2 = (x-y)(x+y)\). Let \(x = 4a^2\) and \(y = 9b^2\). So, \(16a^4 - 81b^4\) becomes \((4a^2 - 9b^2)(4a^2 + 9b^2)\).

Step 3: Factor Further Using Difference of Squares

The term \(4a^2 - 9b^2\) is also a difference of squares: \((2a)^2 - (3b)^2\). Using the difference of squares formula again, it factors into \((2a - 3b)(2a + 3b)\).

Step 4: Combine Factors

Combine all the factors found to express the original expression completely factored: \((2a - 3b)(2a + 3b)(4a^2 + 9b^2)\).

Key Concepts

Difference of Squares

The difference of squares is a concept that helps simplify and factor polynomial expressions. It's all about recognizing a special pattern: a difference between two squared terms. The general formula is \(x^2 - y^2 = (x-y)(x+y)\). This pattern shows how you can express a two-term difference of square numbers as the product of two binomials.

• For instance, the expression \(16a^4 - 81b^4\) is written as \((4a^2)^2 - (9b^2)^2\).
• This means we can apply the difference of squares formula by letting \(x = 4a^2\) and \(y = 9b^2\).

You break down the expression using this formula and it simplifies the factorization process. Recognizing these structures can vastly simplify complex polynomial tasks.

Algebraic Expressions

An algebraic expression is a combination of coefficients, variables, and operations such as addition, subtraction, multiplication, and division. These expressions can range from simple to complex and are fundamental to algebra.

• They can be single terms or involve multiple terms like \(16a^4 - 81b^4\).
• Algebraic expressions need to be manipulated using laws and formulas to simplify or solve them.

Recognizing patterns, such as the difference of squares in an algebraic expression, is crucial. This recognition allows for the application of specific rules that simplify the expressions. Breaking down an expression to its core components enables easier factorization and serves as a building block for solving equations.

Polynomial Equations

Polynomial equations consist of variables raised to whole number powers and coefficients connected by operations. When you factor a polynomial, you are expressing it as a product of simpler polynomials, often by identifying patterns like the difference of squares.

• In our exercise, \(16a^4 - 81b^4\) is a polynomial with two terms.
• Using factorization techniques such as the difference of squares, we simplify it into \((2a - 3b)(2a + 3b)(4a^2 + 9b^2)\).

Factoring polynomial equations can reduce them to simpler forms, making it easier to find their roots or solve them. It's an essential skill in algebra that helps address more complicated problems by reducing them into manageable pieces.