Problem 70 - Elementary and Intermediate Algebra

Question

Factor. $$ 2 p^{4}-32 q^{4} $$

Step-by-Step Solution

Answer

\(2(p - 2q)(p + 2q)(p^2 + 4q^2)\)

Step 1: Identify Common Factors

First, look for any common factors in the expression \(2p^4 - 32q^4\). Both terms have a common factor of 2, so we factor out 2: \(2(p^4 - 16q^4)\).

Step 2: Recognize Difference of Squares

The expression inside the parentheses, \(p^4 - 16q^4\), is a difference of squares since \(16q^4\) can be written as \((4q^2)^2\). Thus, \(p^4 - (4q^2)^2\) can be factored using the formula \(a^2-b^2=(a-b)(a+b)\).

Step 3: Apply Difference of Squares

Now we apply the difference of squares formula: let \(a = p^2\) and \(b = 4q^2\). So, \(p^4 - (4q^2)^2 = (p^2 - 4q^2)(p^2 + 4q^2)\).

Step 4: Recognize Further Factoring Opportunity

Observe that \(p^2 - 4q^2\) is itself a difference of squares because \(4q^2\) is \((2q)^2\). Thus, factor \(p^2 - 4q^2\) again: \(p^2 - (2q)^2 = (p - 2q)(p + 2q)\).

Step 5: Complete Factoring Process

Combine the factored parts. The complete factorization of \(2p^4 - 32q^4\) is therefore \(2(p - 2q)(p + 2q)(p^2 + 4q^2)\).

Key Concepts

Difference of Squares

The difference of squares is a central idea in algebra that can simplify factoring when certain conditions are met. It refers to expressions of the form \(a^2 - b^2\), where two perfect squares are subtracted. This can always be factored into \((a-b)(a+b)\). The power of this method lies in its elegance and simplicity.

In our example, after recognizing that \(p^4 - 16q^4\) is indeed a difference of squares, with \(p^4 = (p^2)^2\) and \(16q^4 = (4q^2)^2\), we can apply the formula directly. This gives us \((p^2 - 4q^2)(p^2 + 4q^2)\).

But don't stop there—sometimes, as in this example, further factorization is possible because \(p^2 - 4q^2\) still fits the pattern of a difference of squares. So it becomes \((p - 2q)(p + 2q)\).

When factoring a difference of squares, always look for opportunities to simplify further by checking each term in the factors whether they themselves can be further factored.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. Understanding how to manipulate these expressions is fundamental to algebra. In the given exercise, \(2p^4 - 32q^4\) is an algebraic expression made up of terms involving powers of variables \(p\) and \(q\) and the coefficient 2.

Learning to identify patterns and applying algebraic identities like factoring difference of squares allows us to break down these expressions into more manageable pieces. This skill is crucial when solving equations, simplifying expressions, or understanding algebra more broadly.

It's also important to understand the role of different operations and terms within an expression. Being able to spot similar patterns of operation, like addition and multiplication of squares, equips students with tools not only to solve current problems but to approach new ones with confidence.

With expressions like these, always start by simplifying them to their simplest components using known algebraic identities and operations.

Greatest Common Factor

Finding the Greatest Common Factor (GCF) is often the very first step when factoring an expression. It refers to the largest factor shared by all the terms in an expression. For the solution \(2p^4 - 32q^4\), the GCF is 2 because both terms, \(2p^4\) and \(32q^4\), are divisible by 2.

Factoring out the GCF makes the rest of the factoring process easier, by reducing the coefficients and simplifying the expression, thus revealing patterns like the difference of squares. After factoring out the GCF, as we did to get \(2(p^4 - 16q^4)\), we're left with a simpler expression within the parentheses to work with.

Recognizing and factoring out the GCF upfront can often lead to mistakes if overlooked, as it might hide certain patterns or lead to more complex numbers than necessary. It is always a good practice to start with this step.

This foundational move not only simplifies the problem but also streamlines subsequent steps in the problem-solving process.