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Chapter 4: Problem 17

\(\frac{18 p^{5}+12 p^{3}-6 p^{2}}{-6 p^{3}}\)

Short Answer

Expert verified
-3 p^{2}-2+ \frac{1}{p}

Step by step solution

01

- Simplify the Fraction

First, observe the expression \(\frac{18 p^{5}+12 p^{3}-6 p^{2}}{-6 p^{3}}\). Rewrite it as a single fraction loop.
02

- Factor the Numerator and Denominator

Factor out the common term in the numerator, which is \(-6 p^{2}\). The expression inside the fraction becomes \(\frac{(-6 p^{2})(-3 p^{3}-2 p+1)}{-6 p^{3}}\).
03

- Cancel Common Terms

Cancel the common terms in the numerator and denominator, which is \(-6 p^{2}\). The fraction reduces to \(\frac{-3 p^{3}-2 p+1}{p}\).
04

- Further Simplify

Divide each term in the numerator by the denominator, p. This results in the simplified expression \(-3 p^{2}-2+ \frac{1}{p}\).

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is a critical skill in algebra. In the given exercise, observe the numerator: \[18 p^{5}+12 p^{3}-6 p^{2}\].\
This polynomial has a common factor, which we identify as \[6 p^{2}\].\
We factor out \[6 p^{2}\] from each term in the numerator:\
- \[18 p^{5} \rightarrow 6 p^{2} \times 3 p^{3}\]
- \[12 p^{3} \rightarrow 6 p^{2} \times 2 p\]
- \[6 p^{2} \rightarrow 6 p^{2} \times 1\]\

This allows us to rewrite the expression as: \[(-6 p^{2})(-3 p^{3} - 2 p + 1)\].\
This factorization is key to further simplifying the fraction.\
It prepares us for the next step, where we'll cancel out common terms.
Canceling Common Terms
Canceling common terms helps simplify expressions by removing identical factors.\
In our exercise, note that both the numerator: \[(-6 p^{2})(-3 p^{3}-2 p+1)\] and the denominator: \[-6 p^{3}\] contain a \[-6 p^{2}\] factor.\

To cancel these terms, divide both out from the numerator and denominator:\
- This gives us \[\frac{(-6 p^{2})(-3 p^{3}-2 p+1)}{-6 p^{2} p}\].\
- Dividing simplifies each part of our expression to: \[\frac{-3 p^{3}-2 p+1}{p}\].\

Always look for these common factors to cancel.\
It's a crucial strategy in simplifying fractions in algebra.
Fraction Division
Understanding fraction division is essential for manipulating algebraic fractions.\
Once our expression is simplified to \[\frac{-3 p^{3}-2 p+1}{p}\], we perform division on each term separately.\

Break it down as follows:\
- \[\frac{-3 p^{3}}{p} \rightarrow -3 p^{2}\]
- \[\frac{-2 p}{p} \rightarrow -2\]
- \[\frac{1}{p} \rightarrow \frac{1}{p}\]

Combining these results, we get the final expression: \[-3 p^{2} - 2 + \frac{1}{p}\].\
Treat each term individually when dividing by a common denominator.\
This technique ensures simplicity and accurate results.

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Most popular questions from this chapter

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications. Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 101 \times 99 $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (2 a+1)^{3} $$

In 2016 , the state of Minnesota had about \(7.33 \times 10^{4}\) farms with an average of \(3.53 \times 10^{2}\) acres per farm. What was the total number of acres devoted to farmland in Minnesota that year? (Data from U.S. Department of Agriculture.)

Light travels at a speed of \(1.86 \times 10^{5} \mathrm{mi}\) per sec. When Venus is \(6.68 \times 10^{7} \mathrm{mi}\) from the sun, how long (in seconds) does it take light to travel from the sun to Venus? (Data from The World Almanac and Book of Facts.)

Graph each equation by completing the table of values. $$ \begin{aligned} &y=(x+3)^{2}\\\ &\begin{array}{c|c|c|c|c|c} x & -5 & -4 & -3 & -2 & -1 \\ \hline y & & & & & \end{array} \end{aligned} $$
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