Problem 64 - Elementary and Intermediate Algebra

Question

Factor. $$ m^{4} n^{4}-16 $$

Step-by-Step Solution

Answer

The fully factored form is \((m n - 2)(m n + 2)(m^2 n^2 + 4)\).

Step 1: Recognize the Expression Form

The given expression is \(m^4 n^4 - 16\). Notice that this is a difference of squares since it can be written as \((m^2 n^2)^2 - 4^2\). A difference of squares has the general form \(a^2 - b^2\), which can be factored as \((a - b)(a + b)\).

Step 2: Apply the Difference of Squares Formula

Rewrite \(m^4 n^4 - 16\) using the difference of squares formula. Set \(a = m^2 n^2\) and \(b = 4\). Therefore, \(a^2 - b^2\) becomes \((m^2 n^2 - 4)(m^2 n^2 + 4)\).

Step 3: Identify Further Factoring Possibilities

The expression \((m^2 n^2 - 4)\) can be further factored, as it is again a difference of squares. Identify this as \((m n)^2 - 2^2\).

Step 4: Factor the Second Difference of Squares

Apply the difference of squares formula to \((m n)^2 - 2^2\). This gives \((m n - 2)(m n + 2)\).

Step 5: Write the Fully Factored Expression

Combine all factored parts from the previous steps. The fully factored form of \(m^4 n^4 - 16\) becomes \((m n - 2)(m n + 2)(m^2 n^2 + 4)\).

Key Concepts

Difference of Squares

The concept of difference of squares is a powerful tool in algebra that allows us to factor certain types of expressions easily. A difference of squares is any expression that can be written in the form \(a^2 - b^2\), where both \(a\) and \(b\) are squared terms. This unique form facilitates easy factoring because it can be expressed as the product of conjugates: \((a - b)(a + b)\).
To identify and work with a difference of squares:

• Look for two squared terms separated by a subtraction sign.
• Express each squared term in the simplest square form.
• Apply the formula \(a^2 - b^2 = (a - b)(a + b)\).

Continuing with the example, \(m^4 n^4 - 16\), recognize \(m^4 n^4\) as \((m^2 n^2)^2\) and \(16\) as \(4^2\). This allows us to use the difference of squares formula to kickstart the factoring process.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules to manipulate these symbols in formulas and equations. It provides the language and tools needed to solve problems systematically and covers a variety of topics such as variables, expressions, equations, and factoring.
When working with algebraic expressions:

• Identify the components of the expression, such as variables and constants.
• Look for opportunities to simplify by factoring, distributing, or combining like terms.
• In exercises like factoring, recognize patterns (e.g., difference of squares) for efficient solutions.

In the example provided, \(m^4 n^4 - 16\) is factored using algebraic concepts to transform it into a more manageable product of terms. Mastery of algebraic rules is essential for recognizing when and how techniques like the difference of squares can be applied.

Quadratic Expressions

Quadratic expressions are polynomial expressions where the highest degree of the variable is two. These expressions can appear in various forms and are often encountered in the context of factoring or solving equations. Typically, they appear as \(ax^2 + bx + c\), but can also be expressed in hidden forms.

• Identify the quadratic nature by looking for squared terms or a specific pattern.
• Utilize techniques such as factoring, completing the square, or the quadratic formula to solve or simplify these expressions.
• Remember that the difference of squares is closely related to quadratics, simplifying a subset of these cases.

In our context with \(m^4 n^4 - 16\), we first reduced it, through the difference of squares, to simpler quadratic forms like \((m n)^2 - 2^2\). Understanding quadratic expressions aids in recognizing and applying suitable algebraic methods to resolve broader polynomial challenges.