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Chapter 6: Problem 42

Write each rational expression in lowest terms. $$ \frac{a^{2}-b^{2}}{a-b} $$

Short Answer

Expert verified
a+b

Step by step solution

01

Identify the problem

The given expression is a rational expression \(\frac{a^{2}-b^{2}}{a-b}\). The task is to simplify it to its lowest terms.
02

Factor the numerator

The numerator is \(a^{2}-b^{2}\). Recognize that \(a^{2}-b^{2}\) is a difference of squares. Use the identity \(x^{2} - y^{2} = (x-y)(x+y)\) to factor it: \(a^{2} - b^{2} = (a-b)(a+b)\).
03

Rewrite the rational expression with factored numerator

Substitute the factored form of the numerator into the original expression: \(\frac{(a-b)(a+b)}{a-b}\).
04

Cancel common factors

In the expression \(\frac{(a-b)(a+b)}{a-b}\), the \(a-b\) terms in the numerator and denominator cancel each other out. Thus, you are left with \(\frac{a+b}{1} = a+b\).

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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 special type of polynomial that we encounter often. It follows the form \(a^{2} - b^{2}\), which can be factored into \((a - b)(a + b)\). This algebraic identity is a result of the product of two conjugates.

To see why this works, expand \((a - b)(a + b)\):

  • First, multiply \(a \cdot a = a^{2}\)
  • Then, multiply \(a \cdot b = ab\)
  • Next, multiply \(-b \cdot a = -ab\)
  • Finally, multiply \(-b \cdot b = -b^{2}\)
Combining these, you get:

a^{2} + ab - ab - b^{2}
The \( + ab \) and \( - ab \) cancel out, leaving \(a^{2} - b^{2}\). Therefore, recognizing a difference of squares allows us to factor polynomials quickly and easily.
factoring
Factoring is breaking down a complicated expression into simpler pieces (factors) which, when multiplied together, give you the original expression. When we notice a difference of squares like \(a^{2} - b^{2}\), we can apply the identity \( (a - b)(a + b)\).

For example, in the given expression\(\frac{a^{2} - b^{2}}{a - b}\)
  • The numerator is \(a^{2} - b^{2}\).
  • We recognize it as a difference of squares.
  • We factor it to \((a-b)(a+b)\).
      • We rewrite the expression then as\frac{(a-b)(a+b)}{a-b}.
      This factored form is essential for the next steps.
canceling common factors
After factoring, we then look for common factors in the numerator and the denominator.
In the expression \( \frac{(a-b)(a+b)}{a-b} \), both the numerator and the denominator have the \(a - b\) term.
We can cancel out these common factors, similar to how we would with numbers.
This simplifies the expression greatly:

  • Cancel the \( (a - b) \) term from the numerator and denominator.
  • We are left with \( \frac{a+b}{1} \).
  • This simplifies to \( a + b \).
Therefore, the simplified form of the given rational expression is \( a + b \).

Remember, when canceling common factors, only do this if the terms multiply together to form the numerator and denominator. Canceling is a powerful tool to greatly simplify rational expressions.

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Most popular questions from this chapter

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