Problem 36 - Elementary and Intermediate Algebra

Question

Factor. \((m+n)^{2}-p^{4}\)

Step-by-Step Solution

Answer

The factored form is \((m+n-p^{2})(m+n+p^{2})\).

Step 1: Recognize the Difference of Squares

The given expression is \((m+n)^{2}-p^{4}\). Notice that this is a difference of squares where \(a = (m+n)\) and \(b = p^{2}\). Thus, it can be expressed as \(a^{2} - b^{2}\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula states that \(a^{2} - b^{2} = (a-b)(a+b)\). Applying this formula, substitute \(a = (m+n)\) and \(b = p^{2}\) to get \((m+n-p^{2})(m+n+p^{2})\).

Step 3: Verify Factorization

To ensure accuracy, distribute each part of the factored expression. Multiply \((m+n-p^{2})\) by \((m+n+p^{2})\) and simplify to show it equals \((m+n)^{2}-p^{4}\). This verifies the factorization is correct.

Key Concepts

Factoring

Factoring in algebra is the process of breaking down a complex expression into simpler components or factors. These factors, when multiplied together, will reconstruct the original expression. In simple terms, factoring can be seen as "unpacking" a mathematical expression. It involves recognizing patterns or using techniques to simplify expressions.

• It helps to break down large expressions into manageable pieces.
• It is essential for solving equations, as it often reveals the roots or solutions of the polynomial.
• Recognizing special identities, like the difference of squares, can speed up the process of factoring.

Factoring is not just about simplifying; it also provides a different perspective on the mathematical problem, making it easier for further manipulation or solving.

Difference of Squares

The difference of squares is a specific pattern in algebra that simplifies the process of factoring. It applies when you have an expression of the form \(a^2 - b^2\), where both terms are perfect squares.
The general strategy for approaching a difference of squares problem is:

• Identify the terms \(a\) and \(b\) such that the expression fits \(a^2 - b^2\).
• Apply the difference of squares formula: \(a^2 - b^2 = (a-b)(a+b)\).

In the given exercise, we recognize \((m+n)^2\) and \(p^4\) as the squares of \((m+n)\) and \(p^2\), respectively. Thus, the expression \((m+n)^2 - p^4\) simplifies to \((m+n-p^2)(m+n+p^2)\).
This technique greatly simplifies certain quadratic expressions, making algebraic manipulations more efficient.

Polynomials

A polynomial is an algebraic expression consisting of variables and coefficients, involving operations of addition, subtraction, and multiplication. Polynomials are foundational elements in algebra that range from simple to complex expressions.
Here's what you need to know about polynomials:

• They can have multiple terms, each consisting of a coefficient and variable raised to a non-negative integer power.
• Terms are usually listed in descending order of their power.
• Polynomial expressions are closed under addition, subtraction, and multiplication, which means performing these operations on polynomials will result in another polynomial.

In the exercise, the expression \((m+n)^2-p^4\) is a polynomial that can be factored due to its special structure as a difference of squares. Understanding how to manipulate and factor polynomials is crucial for solving equations and many other mathematical endeavors.