/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Perform each division. $$ \f... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 30

Perform each division. $$ \frac{-8 k^{4}+12 k^{3}+2 k^{2}-7 k+3}{-2 k} $$

Short Answer

Expert verified
The quotient is \ 4k^3 - 6k^2 - k + 3.5 - \frac{3}{2k}.

Step by step solution

01

Understand the Division

We need to divide the polynomial \(-8k^4 + 12k^3 + 2k^2 - 7k + 3\) by the monomial \(-2k\).
02

Divide Each Term Separately

Each term of the polynomial should be divided by \(-2k\). This means:\(\frac{-8k^4}{-2k}, \frac{12k^3}{-2k}, \frac{2k^2}{-2k}, \frac{-7k}{-2k}, \frac{3}{-2k}\).
03

Perform the Division

Now, we simplify each term individually:1. \( \frac{-8k^4}{-2k} = \frac{8k^4}{2k} = 4k^3 \)2. \( \frac{12k^3}{-2k} = -6k^2 \)3. \( \frac{2k^2}{-2k} = -k \)4. \( \frac{-7k}{-2k} = \frac{7}{2} = 3.5 \)5. \( \frac{3}{-2k} = -\frac{3}{2k} \)
04

Combine the Results

Combine all the simplified terms to get the final expression:\( 4k^3 - 6k^2 - k + 3.5 - \frac{3}{2k} \).

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Key Concepts

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

Monomial Division
Monomial division is simpler than it sounds. When you divide a polynomial by a monomial, you need to divide each term of the polynomial separately by the monomial. For example, in the problem \(\frac{-8k^4 + 12k^3 + 2k^2 - 7k + 3}{-2k}\), you'll divide each term inside the polynomial separately by \(-2k\). This means performing five smaller divisions: \(\frac{-8k^4}{-2k}, \frac{12k^3}{-2k}, \frac{2k^2}{-2k}, \frac{-7k}{-2k}, \frac{3}{-2k}\). Each of these divisions simplifies down to a simpler term. \(\frac{-8k^4}{-2k}\rightarrow 4k^3\).\(\frac{12k^3}{-2k}\rightarrow -6k^2\) and so on. Breaking down the problem makes it more manageable.
Algebraic Expressions
An algebraic expression is a mathematical phrase that includes numbers, variables (like k), and operations (such as addition or division). In the exercise, the expression \(-8k^4 + 12k^3 + 2k^2 - 7k + 3\) contains terms with variables raised to different powers. The goal when dividing or simplifying algebraic expressions is to break them down into more basic parts. When dividing by a monomial, it鈥檚 like separating these terms and simplifying each one individually. This keeps our calculations neat and organized.
Simplifying Terms
Simplifying terms means reducing them to their most basic form. For example, if we have \(\frac{-8k^4}{-2k}\), we simplify it by cancelling out common factors. In this case, dividing -8 by -2 gives 4, and reducing \(\frac{k^4}{k}\) simplifies to \({k^3}\). So, the simplified form of \(\frac{-8k^4}{-2k}\) is 4k^3. Each term is treated the same way: break it down, cancel out common factors, and simplify. After simplifying all terms, we recombine them to get the final simplified expression.

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Most popular questions from this chapter

Find each product. \(-3 k\left(8 k^{2}-12 k+2\right)\)

Graph each equation by completing the table of values. \(y=x^{2}-4\) \(\begin{array}{|c|c|}\hline x & {y} \\ \hline-2 & {} \\ \hline-1 & {} \\\ \hline 0 & {} \\ \hline 1 & {} \\ \hline 2 & {} \\ \hline\end{array}\)

In Objective \(I,\) we showed how \(6^{\circ}\) acts as 1 when it is applied to the product rule, thus motivating the definition of 0 as an exponent. We can also use the quotient rule to motivate this definition. Because \(25=5^{2},\) the expression \(\frac{25}{25}\) can be written as the quotient of powers of \(5 .\) Write the expression in this way.

Evaluate. $$ 38 \div 10 $$

Find each product. \((x+y)^{2}(x-y)^{2}\)
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