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Chapter 5: Problem 33

Factor. See Example 4. $$ (x+y)^{2}-z^{2} $$

Short Answer

Expert verified
The factored form is \((x+y-z)(x+y+z)\).

Step by step solution

01

Identify the Format

The given expression \((x+y)^{2} - z^{2}\) is in the format of \(a^2 - b^2\), where \(a = (x+y)\) and \(b = z\). This is a difference of squares problem.
02

Apply the Difference of Squares Formula

Recall that the difference of squares can be factored using the formula: \(a^2 - b^2 = (a - b)(a + b)\). In this case, substitute \(a = (x+y)\) and \(b = z\) to factor the expression.
03

Substitute and Simplify

After substituting into the formula, we have \((x+y - z)(x+y + z)\). No further simplification is needed because this expression is already in its factored form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
The concept of the "difference of squares" is a key element in algebra, especially when factoring polynomial expressions. A difference of squares is an algebraic expression of the form \(a^2 - b^2\). It represents the difference or subtraction of two squared terms.
To identify a difference of squares, look for an expression where each term is a perfect square and there is a subtraction operation between them. For example, in the expression \((x+y)^2 - z^2\), \((x+y)^2\) and \(z^2\) are both square terms and they are separated by a minus sign.
Understanding the difference of squares helps simplify expressions and solve equations, making it a powerful tool in algebra.
Polynomial Expressions
Polynomial expressions are algebraic expressions that involve sums or differences of terms. Each term consists of a variable raised to a non-negative integer power, multiplied by a coefficient. For example, in the polynomial \((x+y)^2\), the term \((x+y)\) is squared.
Polynomials can have various numbers of terms:
  • Monomials have one term, such as \(3x^2\).
  • Binomials have two terms, like \(x^2 - z^2\).
  • Trinomials contain three terms, such as \(x^2 + 2x + 1\).
Understanding the structure of polynomials and how to manipulate them through operations like addition, subtraction, and factoring is crucial for solving more complex algebraic problems.
Factoring Techniques
Factoring is a process of breaking down an expression into simpler expressions, called factors, that multiply together to give the original expression. This is essential in algebra for simplifying expressions and solving equations.
The "difference of squares" is one specific factoring technique that is particularly simple yet powerful. It involves recognizing the special pattern \(a^2 - b^2\) and factoring it into the product \((a-b)(a+b)\). In our example \((x+y)^2 - z^2\), we see this pattern and apply the technique directly. Here's how it works step by step:
  • Identify the expression as a difference of squares, verifying each term is a perfect square.
  • Use the formula \(a^2 - b^2 = (a-b)(a+b)\).
  • Substitute the appropriate values of \(a\) and \(b\) into the formula \((x+y-z)(x+y+z)\).
By mastering factoring techniques like this one, students can more easily tackle a wide range of algebraic challenges.

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