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Chapter 4: Problem 6

\(\frac{20 x^{4}-25 x^{3}+5 x}{5 x^{2}}\)

Short Answer

Expert verified
4x^2 - 5x + \frac{1}{x}

Step by step solution

01

- Understand the problem

The given expression is a fraction involving polynomials. The goal is to simplify it by dividing the numerator by the denominator.
02

- Factor out the greatest common divisor (GCD)

First, identify the GCD of the terms in the numerator. The GCD is 5x. Rewrite the numerator by factoring out 5x: \[20 x^{4} - 25 x^{3} + 5 x = 5 x (4 x^{3} - 5 x^{2} + 1)\]
03

- Rewrite the expression

Now replace the factored form of the numerator in the original expression: \[\frac{5 x (4 x^3 - 5 x^2 + 1)}{5 x^2}\]
04

- Cancel out common factors

Cancel the common factor of 5x in the numerator and the denominator: \[\frac{5 x (4 x^3 - 5 x^2 + 1)}{5 x^2} = \frac{4 x^3 - 5 x^2 + 1}{x}\]
05

- Simplify the fraction

Finally, simplify each term in the numerator by dividing by x: \[\frac{4 x^3}{x} - \frac{5 x^2}{x} + \frac{1}{x} = 4 x^2 - 5 x + \frac{1}{x}\]

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Key Concepts

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

Greatest Common Divisor
The Greatest Common Divisor (GCD) helps simplify problems involving polynomials. It is the largest factor that divides all terms in the expression equally.
In our exercise, the GCD of the terms in the numerator (20x\textsuperscript{4}, 25x\textsuperscript{3}, 5x) is 5x.
By factoring out this GCD, we can break down the polynomial into simpler terms:
\[20 x^{4} - 25 x^{3} + 5 x = 5 x (4 x^{3} - 5 x^{2} + 1)\]
Factoring out the GCD helps in reducing the complexity of the expression. Always look for common factors in each term when simplifying.
Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into simpler 'factor' expressions that multiply together to make the original polynomial.
In the given exercise, once we factor out the GCD, the polynomial is simpler:
\[20 x^{4} - 25 x^{3} + 5 x = 5 x (4 x^{3} - 5 x^{2} + 1)\]
Factoring can help in various algebraic operations like simplification and solving equations. In our example, factoring the numerator allowed us to cancel out common factors more easily, resulting in a simpler polynomial.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. This usually involves canceling out common factors from the numerator and the denominator.
In our exercize, the simplified fraction is:
\[\frac{5 x (4 x^3 - 5 x^2 + 1)}{5 x^2}\]
We then cancel out the common factor 5x:
\[\frac{4 x^3 - 5 x^2 + 1}{x}\]
Finally, we divide each term by x:
\[4 x^2 - 5 x + \frac{1}{x}\]
This process leads us to a simpler version of the original expression, making it easier to work with.

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