/*! 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 13 Perform each division. $$ \f... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 13

Perform each division. $$ \frac{4 a^{5}-4 a^{2}+8}{4 a} $$

Short Answer

Expert verified
The solution is \(a^{4} - a + \frac{2}{a}\).

Step by step solution

01

Break Down the Fraction

First, break down \(\frac{4 a^{5}-4 a^{2}+8}{4 a}\) into separate fractions: \[ \frac{4 a^{5}-4 a^{2}+8}{4 a} = \frac{4a^{5}}{4a} - \frac{4a^{2}}{4a} + \frac{8}{4a} \]
02

Simplify Each Term Individually

Simplify each fraction: \[\frac{4a^{5}}{4a} = a^{4}\forall \frac{4a^{2}}{4a} = a \frac{8}{4a} = \frac{2}{a} \]
03

Combine the Simplified Terms

Combine the simplified terms: \[ a^{4} - a + \frac{2}{a} \]

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Key Concepts

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

Polynomial Division
Polynomial division involves dividing one polynomial by another. In our example, the polynomial is \(\frac{4a^{5}-4a^{2}+8}{4a}\). We look to simplify each term individually. We achieve this by breaking the polynomial into smaller fractions which can be easily divided. For example, \(\frac{4a^{5}}{4a}\) becomes \(\frac{4a^{5}}{4a}=a^{4}\). This method helps in simplifying complex expressions more manageable.
Fraction Simplification
Fraction simplification is the process of reducing fractions to their simplest form. In this problem, we simplify each individual fraction from our polynomial division. By reducing \(\frac{4a^{5}}{4a}=a^{4}\) and \(\frac{4a^{2}}{4a}=a\), and handling constants like \(\frac{8}{4a}=\frac{2}{a}\), we ensure each part of our fraction is as straightforward as possible. This step-by-step simplification allows us to combine these terms effectively to produce the final simplified expression.
Algebraic Expression
Algebraic expressions are combinations of variables, numbers, and operators. In the given exercise, we are dealing with an algebraic expression of \(\frac{4 a^{5}-4 a^{2}+8}{4 a}\). By breaking it down and simplifying, we translate it into \(\frac{4a^{5}}{4a} - \frac{4a^{2}}{4a} + \frac{8}{4a}\), and further into \('a^{4} - a + \frac{2}{a}\). Understanding and manipulating algebraic expressions are fundamental skills in algebra that aid in solving more complex mathematical problems in the future.

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Most popular questions from this chapter

List all positive integer factors of each number. $$ 48 $$

Simplify: $$ -8(-3 x+7)-4(2 x+3) $$

Use scientific notation to calculate the answer to each problem. (a) The distance to Earth from Pluto is \(4.58 \times 10^{9} \mathrm{km} .\) In April \(1983,\) Pioneer 10 transmitted radio signals from Pluto to Earth at the speed of light, \(3.00 \times 10^{5} \mathrm{km}\) per sec. How long (in seconds) did it take for the signals to reach Earth? (b) How many hours did it take for the signals to reach Earth?

Perform each division using the "long division" process. $$ \frac{5 y^{4}+5 y^{3}+2 y^{2}-y-8}{y+1} $$

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ $$ \text { can be used to perform some multiplication problems. Here are two examples.} $$ $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1^{2} \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 301 \times 299 $$
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