Problem 31 - Elementary and Intermediate Algebra

Question

Factor completely. \(16 a^{4}-81 b^{4}\)

Step-by-Step Solution

Answer

\((4a^2 + 9b^2)(2a + 3b)(2a - 3b)\)

Step 1: Recognize the Expression Type

The given expression is a difference of squares because it can be written as the difference between two terms each raised to an even power: \(16a^4\) and \(81b^4\).

Step 2: Express as Squares

Rewrite \(16a^4\) and \(81b^4\) as squares: \(16a^4 = (4a^2)^2\) and \(81b^4 = (9b^2)^2\). Thus, \(16a^4 - 81b^4 = (4a^2)^2 - (9b^2)^2\).

Step 3: Apply Difference of Squares Formula

The difference of squares formula \(x^2 - y^2 = (x+y)(x-y)\) can be applied here. Let \(x = 4a^2\) and \(y = 9b^2\). The expression can now be factored as \((4a^2 + 9b^2)(4a^2 - 9b^2)\).

Step 4: Factor Further

The second factor \((4a^2 - 9b^2)\) is again a difference of squares. We can write \(4a^2 = (2a)^2\) and \(9b^2 = (3b)^2\). So, \(4a^2 - 9b^2 = (2a)^2 - (3b)^2 = (2a + 3b)(2a - 3b)\).

Step 5: Write the Fully Factored Expression

Substitute back to get the fully factored expression: \((4a^2 + 9b^2)(2a + 3b)(2a - 3b)\).

Key Concepts

Difference of Squares

The difference of squares is a fundamental algebraic concept that allows you to factor expressions composed of two perfect squares separated by a minus sign. When you see an expression like this, the first step is to recognize it as a difference of squares.

The general formula for the difference of squares is \[x^2 - y^2 = (x+y)(x-y)\].This means that any time you have two squares subtracted from each other, you can factor them using this formula.

• Identify terms that are perfect squares, such as \(16a^4\) and \(81b^4\).
• Rewrite each term as a square of another expression, like \((4a^2)^2\) and \((9b^2)^2\).
• Apply the difference of squares formula to break them into two factors.

Recognizing these patterns is key to mastering polynomial factorization, as it simplifies complex expressions quickly.

Factoring Techniques

Factoring techniques are methods used to break down complex algebraic expressions into simpler parts or factors. These techniques are essential in solving equations and simplifying expressions.One key technique is recognizing special forms such as the difference of squares. This involves identifying pairs of terms that can be rewritten using algebraic identities. Another technique involves factoring out the greatest common factor (GCF) from all terms in a polynomial, which simplifies expressions even before using more advanced methods.For instance, in the provided problem, the expression \(16a^4 - 81b^4\) was simplified and broken down into its factorable components. More specifically, the difference of squares method was employed:

• First, identify and factor \((4a^2 + 9b^2)(4a^2 - 9b^2)\).
• Then, notice that \(4a^2 - 9b^2\) is also a difference of squares and needs further factoring.
• Continue expressing each step until the expression is factored completely into \((4a^2 + 9b^2)(2a + 3b)(2a - 3b)\).

Effective factoring requires practice and a solid foundation in recognizing common factorable forms.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. Understanding them is crucial as they form the basis of solving algebraic equations and simplifying expressions.An algebraic expression can involve various operations such as addition, subtraction, multiplication, and division. When working with expressions, the goal is often to simplify or solve them by factoring or combining like terms. In polynomial factorization, you'll often deal with expressions like the given \(16a^4 - 81b^4\), which needs to be broken down into factored parts to simplify or solve further. This process involves:

• Identifying different operations among terms.
• Rewriting terms as squares when possible, as seen with \((4a^2)^2\) and \((9b^2)^2\).
• Applying algebraic identities, like the difference of squares, to facilitate easy factorization.

Being proficient with algebraic expressions, recognizing patterns, and applying factorization techniques allow you to effortlessly simplify complex problems.