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Chapter 4: Problem 14

\(\frac{8 r^{4}-4 r^{3}+6 r^{2}}{2 r}\)

Short Answer

Expert verified
4 r^3 - 2 r^2 + 3 r

Step by step solution

01

Identify the Expression

Firstly, identify the given expression: \ \[\frac{8 r^{4}-4 r^{3}+6 r^{2}}{2 r}\].
02

Factor the Numerator

Factor out the common term in the numerator, which is 2r^2: \ \[8 r^4 - 4 r^3 + 6 r^2 = 2 r^2 (4 r^2 - 2 r + 3).\]
03

Rewrite the Expression

Rewrite the original fraction using the factored form: \ \[\frac{2 r^2 (4 r^2 - 2 r + 3)}{2 r}.\]
04

Simplify the Fraction

Simplify the fraction by canceling common terms. Since 2r is in both the numerator and the denominator, they cancel each other out: \ \[= r (4 r^2 - 2 r + 3).\]
05

Final Simplified Form

The simplified form of the expression is: \ \[4 r^3 - 2 r^2 + 3 r.\]

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Key Concepts

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

Factoring Polynomials
In this exercise, we started by looking at the polynomial in the numerator. Factoring polynomials means finding common terms or expressions that can be pulled out.
For instance, our expression was \(8 r^4 - 4 r^3 + 6 r^2\).
Here, the common term among all the terms is \(2 r^2\).
We factored this out, leading to the expression:
\[2 r^2 (4 r^2 - 2 r + 3).\]
The idea is to make the polynomial easier to handle by expressing it in a product form.
Canceling Common Terms
After factoring the polynomial in the numerator, we rewrote the entire expression. It initially looked like this:
\[\frac{8 r^4 - 4 r^3 + 6 r^2}{2 r}\]
Which we later changed to:
\[\frac{2 r^2 (4 r^2 - 2 r + 3)}{2 r}\]
Our goal here was to cancel common terms found in both the numerator and denominator. The term \(2 r\) appears in both, so we can cancel it out, simplifying our expression further:
\[r (4 r^2 - 2 r + 3).\]
This step helps in reducing the complexity of the expression.
Simplification
The final step in our process is simplification. After canceling the common terms, the expression became:
\[r (4 r^2 - 2 r + 3).\]
When you distribute the \(r\) back in, you get:
\[4 r^3 - 2 r^2 + 3 r.\]
This is our simplified form. Simplification is crucial because it turns a complex expression into a simpler one, making it easier to understand and work with.

Remember, the goal of these steps is to make any rational expression more manageable. Breaking it down into smaller, easier-to-handle pieces is key.

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Most popular questions from this chapter

Perform each indicated operation. \(\left(8 m^{2}-7 m\right)-\left(3 m^{2}+7 m-6\right)\)

Graph each equation by completing the table of values. $$ \begin{aligned} &y=(x-4)^{2}\\\ &\begin{array}{|l|l|l|l|l|l} x & 2 & 3 & 4 & 5 & 6 \\ \hline y & & & & & \end{array} \end{aligned} $$

Perform each indicated operation. \(\left[\left(9 b^{3}-4 b^{2}+3 b+2\right)-\left(-2 b^{3}-3 b^{2}+b\right)\right]-\left(8 b^{3}+6 b+4\right)\)

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Use scientific notation to calculate the result in each expression. Write answers in scientific notation. 78\. \(\frac{0.000015(42,000,000)}{0.000009(0.000005)}\)
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