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Chapter 2: Problem 21

In \(13-24,\) divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{4 b+12}{b} \div(b+3) $$

Short Answer

Expert verified
The simplified form is \(\frac{4}{b}\), undefined for \(b=0\) and \(b=-3\).

Step by step solution

01

Write the Expression as a Fraction Division

The given expression is presented as one fraction divided by another expression. Write it as a fraction division:\[ \frac{4b+12}{b} \div (b+3) \]
02

Rewrite as Multiplication of Reciprocal

To divide by a fraction or expression, multiply by its reciprocal. The reciprocal of \((b+3)\) is \(\frac{1}{b+3}\). Rewrite the expression as:\[ \frac{4b+12}{b} \times \frac{1}{b+3} \]
03

Simplify the Fraction

First, factor the numerator of the first fraction. The expression \(4b+12\) can be factored as \(4(b+3)\). So the fraction becomes:\[ \frac{4(b+3)}{b} \times \frac{1}{b+3} \]
04

Cancel Common Factors

Notice that \((b+3)\) appears in both the numerator and the denominator. Cancel these terms from both the fractions:\[ \frac{4 \cancel{(b+3)}}{b} \times \frac{1}{\cancel{(b+3)}} = \frac{4}{b} \cdot 1 = \frac{4}{b} \]
05

Identify Restrictions on the Variable

The original expression contains two denominators, \(b\) and \((b+3)\). Thus, \(b\) cannot be zero and \(b\) cannot be -3, as these would make the original expression undefined.

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Key Concepts

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

Division of Fractions
Understanding how to divide fractions is a fundamental skill in algebra, and it applies to rational expressions as well. When you see a division sign between two fractions or expressions, think of it as a multiplication problem. This is because dividing by a fraction is the same as multiplying by its reciprocal.
  • The reciprocal of a number is simply flipping the numerator and the denominator. For example, the reciprocal of \(a/b\) is \(b/a\).
  • In our case, we turn \(\frac{4b+12}{b} \, \div \, (b+3)\) into \(\frac{4b+12}{b} \, \times \, \frac{1}{b+3}\).
  • This changing from division to multiplication allows us to more easily simplify the expression.
The idea of reciprocal multiplication really streamlines solving rational expressions, making life easier for you!
Factoring Polynomials
Factoring is an essential tool in simplifying rational expressions. It involves breaking down a polynomial into simpler parts, or factors, that when multiplied together give you the original polynomial.
  • In our exercise, we need to factor the numerator \(4b+12\).
  • This can be rewritten as \(4(b+3)\). We factored out the common factor '4'.
  • Once we've factored, the expression becomes \(\frac{4(b+3)}{b}\).
Factoring can illuminate cancellation opportunities and ultimately simplify the expression. Keep an eye on shared components across the numerators and denominators, like \(b+3\) in this case, which can often be cancelled out to simplify your work.
Undefined Expressions
In mathematics, an undefined expression is one in which the operations can't be performed because they break mathematical rules. These occur typically when a denominator equals zero.
  • In the expression \(\frac{4}{b}\), it's undefined when \(b = 0\), because division by zero is not permitted.
  • Similarly, in the original expression \(b+3\) in the denominator also cannot be zero. Thus, \(b = -3\) would make it undefined.
Knowing these restrictions is critical when you are working with rational expressions or any algebraic expressions. Always check your denominators to avoid getting an undefined result!
Simplifying Expressions
Once you've performed division and factoring, the next logical step is simplification. This involves reducing the expression to its simplest form.
  • With our expression, after multiplying and factoring, we ended with \(\frac{4\cancel{(b+3)}}{b} \times \frac{1}{\cancel{(b+3)}} = \frac{4}{b}\).
  • The simplification occurs by cancelling like terms, such as \(b+3\) in the numerator and denominator.
  • This leaves us with a cleaner and simpler expression that is more manageable.
Simplifying expressions helps in understanding their nature and makes subsequent operations on the expression more straightforward. Always aim for the simplest form!

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Most popular questions from this chapter

Brianna said that \(\frac{3}{x-2}=\frac{5}{x+2}\) is a rational equation but \(\frac{x-2}{3}=\frac{x+2}{5}\) is not. Do you agree with Brianna? Explain why or why not.

Simplify each expression. In each case, list any values of the variables for which the fractions are not defined. \(\left(6+\frac{12}{b}\right) \div\left(3 b-\frac{12}{b}\right)+\frac{b}{2-b}\)

Simplify each complex rational expression. In each case, list any values of the variables for which the fractions are not defined. \(\frac{a-\frac{49}{a}}{a-9+\frac{14}{a}}\)

In \(3-20,\) solve each equation and check. $$ \frac{1}{4} a+8=\frac{1}{2} a $$

When the equation \(2-\frac{3}{b}=\frac{5}{b+2}\) is solved for \(b\) , the solutions are \(-1\) and \(3 .\) Explain why the number line must be separated into five segments by the numbers \(-2,-1,0,\) and 3 in order to check the solution set of the inequality \(2-\frac{3}{b}>\frac{5}{b+2}\)
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