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Chapter 5: Problem 54

Products of Sums and Differences Multiply. $$(x-3)(x+3)$$

Short Answer

Expert verified
(x-3)(x+3) = x^2 - 9

Step by step solution

01

Recognize the Pattern

Notice that the given expression \((x-3)(x+3)\)\ is a product of a sum and a difference, which fits the standard algebraic identity \((a-b)(a+b) = a^2 - b^2\).
02

Identify a and b

Identify the values of \(a\) and \(b\) in the expression. For \(x-3 \) and \(x+3 \), \(a = x\) and \(b = 3\).
03

Apply the Formula

Substitute \(a = x\) and \(b = 3\) into the identity \((a-b)(a+b) = a^2 - b^2\). This gives us \((x-3)(x+3) = x^2 - 3^2\).
04

Simplify the Expression

Simplify the expression further by calculating the square of 3: \(x^2 - 9\).

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Key Concepts

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

Algebraic Identities
Algebraic identities are important tools in mathematics. They are formulas that hold true for all values of the variables involved. One common algebraic identity is the difference of squares formula: e.g. \((a-b)(a+b) = a^2 - b^2\)This identity is useful because it allows us to quickly simplify expressions that match this pattern. Recognizing these patterns takes practice, but once you get the hang of it, you can solve problems more efficiently.
Factoring
Factoring is the process of breaking down an expression into simpler terms, or factors, that when multiplied together give back the original expression.

Why Factoring is Important

Factoring can help simplify complex expressions and solve algebraic equations. For instance:
  • It鈥檚 easier to solve equations when they're factored.
  • Helps in finding roots of polynomials.
In our example problem \((x-3)(x+3)\), we identified that this matches the difference of squares pattern. Here, we were able to factorize the expression quickly using the algebraic identity.
Simplifying Expressions
Simplifying expressions means reducing them to their most basic form. This might involve combining like terms, using algebraic identities, or factoring.

The Process

To simplify \((x-3)(x+3)\):1. Recognize the pattern. \((x-3)(x+3)\) is a difference of squares.2. Use the identity \((a-b)(a+b) = a^2 - b^2\).3. Here, \a=x\ and \b=3\.4. Substitute: \((x)^2 - (3)^2\) = \x^2 - 9\.

Benefits of Simplifying

  • Makes equations easier to solve.
  • Helps in identifying key characteristics of functions.
  • Reduces the chances of making errors in calculations.

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