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Chapter 7: Problem 287

Factor completely. \(121 r^{2}-s^{2}\)

Short Answer

Expert verified
\((11r + s)(11r - s)\)

Step by step solution

01

Identify the form of the expression

The given expression is \(121r^{2} - s^{2}\). Notice that it resembles the difference of squares form \(a^{2} - b^{2}\). The formula for factoring difference of squares is \(a^{2} - b^{2} = (a + b)(a - b)\).
02

Rewrite each term as a square

Identify the squares in the terms: \(121r^{2} = (11r)^{2}\) and \(s^{2} = (s)^{2}\). Thus, the expression can be written as \((11r)^{2} - (s)^{2}\).
03

Apply the difference of squares formula

Using the difference of squares formula, substitute \(a = 11r\) and \(b = s\). The factored form of the expression is \((11r + s)(11r - s)\).

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Key Concepts

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

difference of squares
The difference of squares is a specific algebraic identity where two perfect squares are subtracted. It's important because it allows complex expressions to be simplified easily. The formula is \[a^2 - b^2 = (a + b)(a - b)\]. This means any expression that fits this pattern can be factored into two binomials. For example, in the original exercise, \(121r^2 - s^2\) can be rewritten by recognizing that \(121r^2 = (11r)^2\) and \(s^2 = (s)^2\). Thus, you can apply the difference of squares formula: (11r)^{2} - (s)^{2} = (11r + s)(11r - s). Understanding how to identify and use this formula is crucial in simplifying many algebraic expressions.
factoring polynomials
Factoring polynomials involves breaking down a polynomial into simpler expressions (factors) that when multiplied together give back the original polynomial. It's like reversing multiplication. In the given exercise with the expression \(121r^{2} - s^{2}\), we see that this is a polynomial in two variables. By recognizing this as a difference of squares, we can factor it as (11r + s)(11r - s). Here are the steps to factor a polynomial:
  • Identify the form (e.g., difference of squares, trinomial, etc.)
  • Rewrite the terms as squares or other identifiable forms
  • Apply the appropriate factoring formula
  • Simplify, if necessary
This technique is widely applicable in different areas of algebra, from solving equations to simplifying expressions.
algebraic expressions
An algebraic expression is a combination of variables, numbers, and operations. For example, \(121r^{2} - s^{2}\) is an algebraic expression that includes terms, coefficients, and a subtraction operation. Understanding how to manipulate these expressions is key in algebra. Here are some helpful tips:
  • Identify and group similar terms
  • Use known formulas (like the difference of squares) to simplify
  • Check your work by expanding your factors back out
Mastering algebraic expressions helps in solving equations, graphing, and many real-world applications where relationships are described mathematically. The key is to practice recognizing patterns and applying the correct simplification techniques.

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