Problem 47 - A Graphical Approach to College Algebra

Question

Factor each difference of squares completely. $$p^{4}-625$$

Step-by-Step Solution

Answer

\((p - 5)(p + 5)(p^2 + 25)\)

Step 1: Identify the difference of squares pattern

The difference of squares pattern follows the formula \(a^2 - b^2 = (a-b)(a+b)\). In this expression, we see \(p^4\) as a perfect square, which is \((p^2)^2\), and 625 as another perfect square, which is \((25)^2\). Thus, the expression can be identified as \((p^2)^2 - (25)^2\).

Step 2: Apply the difference of squares formula

Using the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\), substitute \(a = p^2\) and \(b = 25\). Thus, the expression becomes \((p^2 - 25)(p^2 + 25)\).

Step 3: Factor further if possible

Notice that \(p^2 - 25\) can further be factored as it's also a difference of squares: \(p^2 - 25 = (p - 5)(p + 5)\). However, the term \(p^2 + 25\) is not factorable in terms of real numbers. Therefore, the completely factored form is \((p - 5)(p + 5)(p^2 + 25)\).

Key Concepts

Difference of Squares

The difference of squares refers to an algebraic expression where two terms are perfect squares separated by a subtraction sign: \(a^2 - b^2\). This pattern is unique because it can be factored as \((a-b)(a+b)\). This simplification relies on the special binomial product formula that results in the subtraction of two squares. It is a handy technique when working with polynomials, making it easier to break down more complex expressions.

Consider the expression \(p^4 - 625\). Notice first that \(p^4\) is the square of \(p^2\), or \((p^2)^2\). Similarly, 625 is the square of 25, or \((25)^2\). Using the formula, we can rewrite the expression as \((p^2 - 25)(p^2 + 25)\). This technique simplifies polynomials into a product of two terms, significantly easing more advanced computations or integrations.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. These expressions are the core components in algebra and can represent a wide range of problems.

Understanding the parts of an algebraic expression helps you manipulate and solve problems effectively. For instance, in the expression \(p^4 - 625\), \(p^4\) and 625 are terms connected by a subtraction sign. Both are also perfect squares, facilitating the application of the difference of squares technique.

• //**Terms**: Parts of an algebraic expression separated by plus or minus signs.
• //**Coefficients**: Numbers that multiply variables.
• //**Variables**: Symbols, typically letters, representing unknown values.
• //**Constants**: Fixed values that do not change.

Recognizing these parts transforms seemingly complex expressions into solvable components, allowing for efficient solutions.

Polynomial Factoring Techniques

Factoring polynomials is a crucial skill in algebra that involves expressing a polynomial as a product of simpler polynomials. It's like breaking down a number into its prime factors but applied to polynomials. Different techniques are used depending on the type of polynomial you need to factor.

There are several polynomial factoring techniques used in algebra:

• //**Greatest Common Factor (GCF)**: Find the largest factor shared by all terms in the polynomial.
• //**Difference of Squares**: Useful for expressions like \(a^2 - b^2\) allows simplification into \((a-b)(a+b)\).
• //**Trinomials**: Factoring trinomials into two binomials like \(x^2 + bx + c\).
• //**Grouping**: For polynomials with four or more terms, factoring by grouping may be effective.

In the original exercise, we applied the difference of squares technique first to write \(p^4 - 625\) as \((p^2 - 25)(p^2 + 25)\). Then, recognizing \(p^2 - 25\) as another difference of squares, we further factored it into \((p - 5)(p + 5)\). Understanding these techniques is essential for simplifying complex polynomials and solving algebraic equations easily.