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Chapter 6: Problem 39

Write each rational expression in lowest terms. $$ \frac{3 z^{2}+z}{18 z+6} $$

Short Answer

Expert verified
\(\frac{z}{6}\)

Step by step solution

01

Factor the Numerator

To simplify the rational expression, start by factoring the numerator. The numerator is given as \(3z^2 + z\). Factor out the greatest common factor (GCF), which is \(z\). This gives: \[3z^2 + z = z(3z + 1)\]
02

Factor the Denominator

Next, factor the denominator. The denominator is \(18z + 6\). Factor out the GCF of \(18z + 6\), which is \(6\). This gives: \[18z + 6 = 6(3z + 1)\]
03

Simplify the Rational Expression

After factoring, the rational expression now looks like this: \[\frac{z(3z + 1)}{6(3z + 1)}\]. The common factor \(3z + 1\) appears in both the numerator and the denominator. Cancel out \(3z + 1\) from both. This leaves: \[\frac{z}{6}\]

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is a crucial skill in algebra. It involves breaking down a polynomial into simpler parts called factors. For example, consider the numerator in our exercise, \(3z^2 + z\). The goal is to express this polynomial as a product of its factors.
First, identify the greatest common factor (GCF) of all terms in the polynomial. Here, both terms, \(3z^2\) and \(z\), have \(z\) as a common factor. So, we factor out \(z\):
\[3z^2 + z = z(3z + 1)\]
This breaks the numerator into two simpler parts: \(z\) and \(3z + 1\). Factoring like this is often the first step in simplifying more complex algebraic expressions.

Remember:
  • Always look for a common factor in all terms.
  • Rewrite the polynomial as a product of this factor and the remaining simplified polynomial.
Greatest Common Factor (GCF)
The greatest common factor (GCF) is the largest factor that two or more numbers or terms have in common. Identifying the GCF is a key step in simplifying polynomials and rational expressions.
In our example, we had to simplify both the numerator and the denominator. Let鈥檚 break it down:
For the numerator \(3z^2 + z\), the GCF is \(z\) because it is the highest factor present in both terms.
For the denominator \(18z + 6\), the GCF is \(6\) because it is the largest number that can divide both 18 and 6 completely.
Once identified, factoring out the GCF makes it easier to see and cancel common elements, simplifying the expression.

Steps to find the GCF:
  • List the factors of each term.
  • Find the largest factor common to all terms.
Canceling Common Factors
After factoring the numerator and the denominator, we often find common factors that can be canceled out. In our simplified expression:
\[ \frac{z(3z + 1)}{6(3z + 1)}\]
Both the numerator and the denominator contain the term \(3z + 1\). This common factor can be canceled out, leaving us with:
\[ \frac{z}{6} \]
Canceling common factors helps reduce rational expressions to their lowest terms, making them simpler to work with.

Key points to remember:
  • Factor both the numerator and the denominator completely.
  • Identify and cancel out identical factors in both the numerator and the denominator.
  • Rewrite the rational expression in its simplest form.

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Most popular questions from this chapter

Which rational expression can be simplified? A. \(\frac{x^{2}+2}{x^{2}}\) B. \(\frac{x^{2}+2}{2}\) C. \(\frac{x^{2}+y^{2}}{y^{2}}\) D. \(\frac{x^{2}-5 x}{x}\)

Identify the two terms in the numerator and the two terms in the denominator of the rational expression \(\frac{x^{2}+4 x}{x+4},\) and write it in lowest terms.

Add or subtract as indicated. $$\frac{8}{(3 r-1)^{2}}+\frac{2}{3 r-1}-6$$

A student was asked to simplify the rational expression $$ \frac{2 x+y}{4 x+2 y} $$ and wrote the incorrect answer \(\frac{x+y}{4 x+y}\) WHAT WENT WRONG? What is the correct answer?

Add or subtract as indicated. $$\frac{2 x+4}{x+3}+\frac{3}{x}-\frac{6}{x^{2}+3 x}$$
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