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Chapter 7: Problem 24

In the following exercises, simplify each rational expression. $$ \frac{4 v-32}{64-v^{2}} $$

Short Answer

Expert verified
-\frac{4}{v + 8}

Step by step solution

01

Factor the Numerator

Identify and factor out the greatest common factor (GCF) of the numerator. The numerator is given by \(4v - 32\). The GCF of 4 and 32 is 4. Factoring out the 4 gives us: \[ 4(v - 8) \]
02

Factor the Denominator

The denominator is \(64 - v^2\). Notice that this is a difference of squares. Recall that the difference of squares can be factored using the identity \(a^2 - b^2 = (a - b)(a + b)\), where \(a = 8\) and \(b = v\). Therefore, we can rewrite the denominator as: \[ (8 - v)(8 + v) \]
03

Simplify the Expression

Substitute the factored forms of the numerator and denominator into the rational expression: \[ \frac{4(v - 8)}{(8 - v)(8 + v)} \] Observe that \(8 - v\) can be rewritten as \(-(v - 8)\). So, the expression can be further simplified: \[ \frac{4(v - 8)}{-(v - 8)(8 + v)} = -\frac{4}{8 + v} \] if \(v eq 8\) and \(v eq -8\).

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Key Concepts

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

Greatest Common Factor (GCF)
To simplify rational expressions, start by factoring out the greatest common factor (GCF) from the numerator and denominator.

The GCF is the largest number or expression that evenly divides all terms. In the exercise, the numerator is \( 4v-32 \). Both 4 and 32 have a common factor of 4.

Factoring out the 4, we rewrite \( 4v-32 \) as \( 4(v-8) \).

Always search for the GCF first, as this simplifies the expression and makes further factoring easier.
Difference of Squares
The denominator in our problem, \( 64-v^2 \), is an example of a difference of squares. This form is characterized by an equation that can be represented as \( a^2 - b^2 \). For our purposes:

- \( a^2 = 64 \to a = 8 \)
- \( b^2 = v^2 \to b = v \)

The difference of squares can be factored using the formula: $$ a^2 - b^2 = (a - b)(a + b) $$.

Therefore, we rewrite \( 64-v^2 \) as \( (8-v)(8+v) \).

Identifying and using the difference of squares can greatly simplify the rational expression.
Factoring Polynomials
Factoring polynomials involves breaking them down into simpler components, often in parentheses.

For the numerator, \( 4v-32 \), we factored out the GCF giving \( 4(v-8) \).

For the denominator, \( 64-v^2 \), we used the difference of squares: \( (8-v)(8+v) \).

Factoring polynomials simplifies rational expressions by reducing them to their simplest forms.

Remember to always look for common factors and special products like the difference of squares.
Rational Expressions
A rational expression is a fraction where the numerator and/or denominator are polynomials. Simplifying these expressions involves factoring and reducing common terms.

In our problem, we started by factoring both numerator and denominator:
$$ \frac{4(v-8)}{(8-v)(8+v)} $$.

Next, notice that \( 8-v \) is the negative of \( v-8 \). Thus: $$ \frac{4(v-8)}{-(v-8)(8+v)} = -\frac{4}{8+v} $$.

Simplifying rational expressions help in solving more complex algebraic problems because they are easier to work with in this simplified form.

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Most popular questions from this chapter

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