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Chapter 7: Problem 34

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{z^{2}-3 z+2}{z^{2}+4 z+3} \div \frac{z-1}{z+1}$$

Short Answer

Expert verified
\[\frac{z-2}{z+3}\].

Step by step solution

01

- Rewrite the division as multiplication

Recall that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, \(\frac{z^{2}-3 z+2}{z^{2}+4 z+3} \times \frac{z+1}{z-1}\).
02

- Factorize the numerators and denominators

Factorize the quadratic expressions. For \(z^{2}-3z+2\), you get \((z-1)(z-2)\). For \(z^{2}+4z+3\), you get \((z+1)(z+3)\). The expression becomes:\[\frac{(z-1)(z-2)}{(z+1)(z+3)} \times \frac{z+1}{z-1}\].
03

- Simplify the expression

In the multiplication of the fractions, cancel out common terms in the numerator and denominator:\[\frac{(z-1)(z-2)}{(z+1)(z+3)} \times \frac{z+1}{z-1} = \frac{z-2}{z+3}\].
04

- Write the final answer in lowest terms

After simplifying, the final expression is: \[\frac{z-2}{z+3}\]. Ensure that it is in its simplest form and confirm there are no more common factors.

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Key Concepts

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

Factoring Quadratics
Quadratics are algebraic expressions of the form \( ax^2 + bx + c \). Factoring quadratics means expressing this form as a product of two linear expressions. Here's how you do it:
  • Find two numbers that multiply to \( a \times c \) and add up to \ b \.
  • Break the middle term \ bx \ into two terms using these numbers.
  • Factor by grouping.

For example, in \( z^2 - 3z + 2 \), the coefficients are \( a=1, b=-3, \) and \ c=2 \. We need two numbers that multiply to \(1 \times 2 \) (which is 2) and add up to -3. These numbers are -1 and -2. So we rewrite \(-3z \) as \(-1z - 2z\) and factor by grouping:
\[ z^2 - z - 2z + 2 = z(z - 1) - 2(z - 1) = (z - 1)(z - 2) \] Similarly, factor \( z^2 + 4z + 3 \) into \((z + 1)(z + 3)\).
Multiplying Fractions
When multiplying fractions, the general rule is to multiply the numerators together and the denominators together. Here鈥檚 the step-by-step process:
  • Multiply the numerators (the top numbers) of the fractions.
  • Multiply the denominators (the bottom numbers) of the fractions.
  • Simplify the resulting fraction if possible by canceling common factors.

In the exercise given, we need to multiply the expression \( \frac{(z-1)(z-2)}{(z+1)(z+3)} \) by the reciprocal of the second fraction. The reciprocal of \( \frac{z-1}{z+1} \) is \( \frac{z+1}{z-1} \). Thus, we multiply:
\[ \frac{(z-1)(z-2)}{(z+1)(z+3)} \times \frac{z+1}{z-1} \] After multiplying, cancel common factors on the numerator and denominator to simplify.
Simplifying Algebraic Expressions
Simplifying algebraic expressions aims to reduce them to their simplest form. Follow these steps:
  • Factorize both the numerator and the denominator (if possible).
  • Cancel out any common factors present in both the numerator and the denominator.
  • Write down the simplified expression.

In terms of our example, after multiplying fractions, we have:
\[ \frac{(z-1)(z-2)}{(z+1)(z+3)} \times \frac{z+1}{z-1} = \frac{(z-1)(z-2)(z+1)}{(z+1)(z+3)(z-1)} \] Here, \( (z-1) \) and \( (z+1) \) appear in both the numerator and the denominator. Cancel these out to get:
\[ \frac{z-2}{z+3} \] Always ensure that the final result is in its simplest form and check for any possible errors in cancellation.

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Most popular questions from this chapter

Determine whether each equation represents direct or inverse variation. $$ y=\frac{3}{x} $$

Find each power. $$ 3.5^{2} $$

Solve each equation for \(k\) $$ 25=9 k $$

Rewrite each rational expression with the indicated denominator. $$ \frac{36 r}{r^{2}-r-6}=\frac{?}{(r-3)(r+2)(r+1)} $$

Add or subtract as indicated. Write each answer in lowest terms. $$ \frac{2}{3}+\frac{8}{27} $$
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