Chapter 7: Q. 68 (page 653)

For the following exercises, find $R (x) = \frac{f (x)}{g (x)}$ where $f (x)$ and $g (x)$ are given.
role="math" localid="1646212760399" $f (x) = \frac{24 x^{2}}{2 x + 8} g (x) = \frac{4 x^{3} + 28 x^{2}}{x^{2} + 11 x + 28}$

Short Answer

Expert verified

The required rational expression is $\frac{3 (x + 4)}{(x - 4)}$ .

Step by step solution

Step 1. Given information

The given expressions are $f (x) = \frac{24 x^{2}}{2 x + 8} g (x) = \frac{4 x^{3} + 28 x^{2}}{x^{2} + 11 x + 28}$

Step 2. Rewrite the Expression as a Product

Substitute the given values into $R (x) = \frac{f (x)}{g (x)}$ .

$R (x) = \frac{\frac{24 x^{2}}{2 x - 8}}{\frac{4 x^{3} + 28 x^{2}}{x^{2} + 11 x + 28}}$

Write the expression of $R (x)$ as the product of numerator and the reciprocal of denominator.

$R (x) = \frac{24 x^{2}}{2 x - 8} \times \frac{x^{2} + 11 x + 28}{4 x^{3} + 28 x^{2}}$

Step 3. Factor the Numerator and Denomiantor

Express the numerator and denominator of the obtained expression as factors.

$R (x) = \frac{6 \times 4 \times x^{2}}{2 (x - 4)} \times \frac{(x + 7) (x + 4)}{4 \times x^{2} \times (x + 7)}$

Step 4. Cancel out the Common Factors

Simplify the obtained expression by canceling out the common factors.

$\begin{array}{rcl} R (x) & = & \frac{2 \times 3 \times 4 \times x^{2}}{2 (x - 4)} \times \frac{(x + 7) (x + 4)}{4 \times x^{2} \times (x + 7)} \\ = & \frac{3}{x - 4} \times x + 4 \\ = & \frac{3 (x + 4)}{(x - 4)} \end{array}$

So, the simplified expression is $R (x) = \frac{3 (x + 4)}{(x - 4)}$ .

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.

For the following exercises, find $R (x) = \frac{f (x)}{g (x)}$ where $f (x)$ and $g (x)$ are given.
role="math" localid="1646212760399" $f (x) = \frac{24 x^{2}}{2 x + 8} g (x) = \frac{4 x^{3} + 28 x^{2}}{x^{2} + 11 x + 28}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Rewrite the Expression as a Product

Step 3. Factor the Numerator and Denomiantor

Step 4. Cancel out the Common Factors

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Theoretical and Mathematical Physics

Decision Maths

Applied Mathematics

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

For the following exercises, find R(x)=f(x)g(x)where f(x)and g(x)are given.role="math" localid="1646212760399" f(x)=24x22x+8g(x)=4x3+28x2x2+11x+28

Short Answer

Step by step solution

Step 1. Given information

Step 2. Rewrite the Expression as a Product

Step 3. Factor the Numerator and Denomiantor

Step 4. Cancel out the Common Factors

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Theoretical and Mathematical Physics

Decision Maths

Applied Mathematics

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

For the following exercises, find $R (x) = \frac{f (x)}{g (x)}$ where $f (x)$ and $g (x)$ are given.
role="math" localid="1646212760399" $f (x) = \frac{24 x^{2}}{2 x + 8} g (x) = \frac{4 x^{3} + 28 x^{2}}{x^{2} + 11 x + 28}$