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Chapter 7: Problem 60

Write each rational expression in lowest terms. $$ \frac{p^{2}+q^{2}}{p^{2}-q^{2}} $$

Short Answer

Expert verified
The expression is in its simplest form as \( \frac{p^{2} + q^{2}}{(p + q)(p - q)} \).

Step by step solution

01

Identify the numerator and the denominator

Start by recognizing the parts of the rational expression. The numerator is \( p^{2} + q^{2} \) and the denominator is \( p^{2} - q^{2} \).
02

Factor the numerator and the denominator, if possible

Check if the numerator \( p^{2} + q^{2} \) or the denominator \( p^{2} - q^{2} \) can be factored. Notice that \( p^{2} + q^{2} \) is a sum of squares which cannot be factored further in real numbers. The denominator \( p^{2} - q^{2} \) is a difference of squares and can be factored into \( (p + q)(p - q) \).
03

Rewrite the expression with the factored form of the denominator

Incorporate the factoring done in the previous step. The rational expression is now \( \frac{p^{2} + q^{2}}{(p + q)(p - q)} \).
04

Check for common factors

Look for any common factors in the numerator and the denominator that can be canceled out. Since \( p^{2} + q^{2} \) and \( (p + q)(p - q) \) do not share any common factors (other than 1), no further cancellation can be done.
05

Conclusion

The rational expression \( \frac{p^{2} + q^{2}}{p^{2} - q^{2}} \) is already in its simplest 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 topic in algebra. It's all about recognizing a specific kind of polynomial that can be factored in a special way. This polynomial takes the form: \[ a^2 - b^2 \]. Here鈥檚 how it works:
  • Identify two squares being subtracted.
  • Use the formula: \[ a^2 - b^2 = (a + b)(a - b) \].
For example, \[ p^2 - q^2 \] can be rewritten as \[ (p + q)(p - q) \]. This makes math problems simpler, especially when simplifying rational expressions. In our exercise, this factoring transformed the denominator \( p^2 - q^2 \) into \( (p+q)(p-q) \). Remember, identifying a 'difference of squares' in your equations can greatly simplify your work.
factoring polynomials
When simplifying rational expressions, 'factoring polynomials' is essential. Factoring breaks down a polynomial into simpler polynomials that multiply together to give the original polynomial. The steps include:
  • Identifying a pattern (like difference of squares, trinomials, or common factors).
  • Using the appropriate factoring formula.
For example, in our exercise, the denominator \( p^2 - q^2 \) was recognized as a difference of squares, so we factored it into \( (p + q)(p - q) \). However, the numerator \( p^2 + q^2 \) is a sum of squares and cannot be factored further in real numbers. Understanding how to factor polynomials correctly is key to simplifying expressions and solving equations.
lowest terms
Writing a rational expression in 'lowest terms' means simplifying it as much as possible. This involves canceling out common factors in the numerator and the denominator. The key steps are:
  • Factor both the numerator and the denominator completely.
  • Look for and cancel any common factors.
  • If no common factors exist, the expression is already in its simplest form.
In our exercise, the rational expression \( \frac{p^2 + q^2}{p^2 - q^2} \) was simplified by first factoring the denominator to \( (p + q)(p - q) \). But since the numerator and the newly factored denominator had no common factors, no further simplification was possible. Thus, the expression was already in its lowest terms. Making sure your rational expressions are in their simplest form can often make solving problems easier.

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Most popular questions from this chapter

Multiply or divide as indicated. $$ \frac{a^{4}}{5 b^{2}} \cdot \frac{25 b^{4}}{a^{3}} $$

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