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Chapter 6: Problem 77

Multiply or divide as indicated. $$ \left(z^{2}-1\right) \cdot \frac{1}{1-z} $$

Short Answer

Expert verified
-z - 1

Step by step solution

01

Factorize the numerator

Observe the term \((z^2 - 1)\). This is a difference of squares which can be factorized as \(z^2 - 1 = (z - 1)(z + 1)\). Rewrite the expression using this factorization.
02

Write the expression with factored numerator

Substitute the factorized form into the expression: \((z-1)(z+1) \cdot \frac{1}{1 - z}\).
03

Simplify the denominator

Notice that \(1 - z\) can be written as \(-(z - 1)\). Substitute this into the expression: \( (z-1)(z+1) \cdot \frac{1}{-(z - 1)} \).
04

Simplify the expression

Cancel out the common terms \(z - 1\) in the numerator and the denominator: \( (z-1)(z+1) \cdot \frac{1}{-(z - 1)} = (z + 1) \cdot \frac{1}{-1} \).
05

Final simplification

Simplify \( (z + 1) \cdot \frac{1}{-1} = -(z + 1) \). The final expression is \ -z - 1 \.

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Key Concepts

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

Difference of Squares
In algebra, the *difference of squares* is a very common concept. It refers to an expression that can be written in the form \( a^2 - b^2 \). This is a special factorization pattern because this difference can always be factored into two binomials. For example, the expression \( z^2 - 1 \) can be written as \( (z - 1)(z + 1) \).This pattern works because \( a^2 - b^2 \) always equals \((a - b)(a + b)\).This is a handy tool for simplifying algebraic expressions.
Factorization
*Factorization* is the process of breaking down an expression into its simplest parts (factors). Factorizing can involve numbers, variables, or a combination of both. With our example, the expression \( z^2 - 1 \) was factorized into \( (z - 1)(z + 1) \).This is crucial because simplifying expressions often starts with factorizing. When you see an expression you suspect can be factored, look for patterns like the difference of squares. Some tips:
  • Look for common factors first.
  • Check for special patterns (e.g., difference of squares, perfect square trinomials).
  • Factor step-by-step and simplify as you go.
This process makes complex expressions easier to handle.
Simplifying Rational Expressions
*Simplifying rational expressions* involves reducing fractions where both the numerator and the denominator are polynomials. The main steps include:
  • Factorizing both the numerator and the denominator.
  • Canceling out common factors in the numerator and the denominator.
In our example, the expression \( (z - 1)(z + 1) \frac{1}{1 - z} \) can simplify by noticing that \(1 - z \) can be rewritten as \( -(z - 1) \). So, we substitute and simplify by canceling common terms. After simplifying fraction expressions, always check for restrictions, like values that make the denominator zero, which are not allowed.
Algebraic Manipulation
*Algebraic manipulation* entails performing operations to rewrite expressions in a more desirable form. You may add, subtract, multiply, divide, or factorize terms in an expression. In this example, a couple of key manipulations include:
  • Rewriting \(1 - z \) as \(-(z - 1) \), a useful step for simplification.
  • Canceling common factors such as \(z - 1 \) in both the numerator and the denominator.
Algebraic manipulation is essential for solving algebra problems, making expressions easier to work with, and arriving at simplified final forms.
Keep these principles in mind, and you鈥檒l find handling algebraic expressions more intuitive over time.

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Most popular questions from this chapter

Write each rational expression in lowest terms. $$ \frac{3 x+7}{3} $$

Add or subtract as indicated. $$\frac{15}{y^{2}+3 y}+\frac{2}{y}+\frac{5}{y+3}$$

Multiply or divide as indicated. $$ \frac{(x+2)(x+1)}{(x+3)(x-2)} \cdot \frac{(x+3)(x+4)}{(x+2)(x+1)} $$

As review, multiply or divide the rational numbers as indicated. Write answers in lowest terms. $$ \frac{5}{6} \div \frac{14}{15} $$

Write each rational expression in lowest terms. $$ \frac{(2 x+7)(x-1)}{(2 x+3)(2 x+7)} $$
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