Chapter 7: Q. 128 (page 668)

In the following exercises, perform indicated operations.
$\frac{7 q + 8}{q^{2} - 2 q - 24} - \frac{q + 2}{4 - q}$

Short Answer

Expert verified

The required expression is $\begin{array}{rcl} \frac{- (q^{2} - 9 q - 32)}{(q - 6) (q + 4)} \end{array}$ .

Step by step solution

Step 1. Given information

The given expression is $\frac{7 q + 8}{q^{2} - 2 q - 24} - \frac{q + 2}{4 - q}$ .

Step 2. Find LCD

Factor the denominator of first term.

$q^{2} - 2 q - 24 = (q - 6) (q + 4)$

As the denominators have no common term, the LCD is $(q - 6) (q + 4) (4 - q)$

Step 3. Express with LCD

Express the given expression with LCD as denominators of both the terms.

$\begin{array}{rcl} \frac{7 q + 8}{q^{2} - 2 q - 24} - \frac{q + 2}{4 - q} & = & \frac{(7 q + 8) (4 - q)}{(q - 6) (q + 4) (4 - q)} - \frac{(q + 2) (q - 6) (q + 4)}{(q - 6) (q + 4) (4 - q)} \\ = & \frac{- 7 q^{2} + 20 q + 32}{(q - 6) (q + 4) (4 - q)} - \frac{- q^{3} + 6 q^{2} + 16 q - 96}{(q - 6) (q + 4) (4 - q)} \end{array}$

Step 4. Simplify

As the denominator is the same for both the terms, perform subtraction in the numerator and place it over the common denominator.

$\begin{array}{rcl} \frac{- 7 q^{2} + 20 q + 32 + q^{3} - 6 q^{2} - 16 q + 96}{(q - 6) (q + 4) (4 - q)} & = & \frac{q^{3} - 13 q^{2} + 4 q + 128}{(q - 6) (q + 4) (4 - q)} \\ = & \frac{(q - 4) (q^{2} - 9 q - 32)}{(q - 6) (q + 4) (4 - q)} \\ = & \frac{- (q^{2} - 9 q - 32)}{(q - 6) (q + 4)} \end{array}$

So, the required expression is $\begin{array}{rcl} \frac{- (q^{2} - 9 q - 32)}{(q - 6) (q + 4)} \end{array}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In the following exercises, perform indicated operations.
$\frac{7 q + 8}{q^{2} - 2 q - 24} - \frac{q + 2}{4 - q}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find LCD

Step 3. Express with LCD

Step 4. Simplify

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

In the following exercises, perform indicated operations.7q+8q2-2q-24-q+24-q

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find LCD

Step 3. Express with LCD

Step 4. Simplify

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

In the following exercises, perform indicated operations.
$\frac{7 q + 8}{q^{2} - 2 q - 24} - \frac{q + 2}{4 - q}$