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Chapter 4: Problem 10

Divide. $$ \frac{64 x^{3}-72 x^{2}+12 x}{8 x^{3}} $$

Short Answer

Expert verified
8 - \frac{9}{x} + \frac{3}{2 x^{2}}

Step by step solution

01

- Separate the terms

Divide each term of the numerator by the denominator separately:\[ \frac{64 x^{3}}{8 x^{3}} - \frac{72 x^{2}}{8 x^{3}} + \frac{12 x}{8 x^{3}} \]
02

- Simplify each fraction

Simplify each fraction individually:\[ \frac{64 x^{3}}{8 x^{3}} = 8 \]\[ \frac{72 x^{2}}{8 x^{3}} = 9 x^{-1} = \frac{9}{x} \]\[ \frac{12 x}{8 x^{3}} = \frac{3}{2} x^{-2} = \frac{3}{2 x^{2}} \]
03

- Combine the simplified terms

Combine all the simplified terms back together:\[ 8 - \frac{9}{x} + \frac{3}{2 x^{2}} \]

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Key Concepts

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

Simplifying Fractions
To simplify fractions, we break down both the numerator and the denominator into their factors and cancel out common factors. This process makes fractions easier to work with.

In the provided exercise, the original fraction is \(\frac{64 x^{3}-72 x^{2}+12 x}{8 x^{3}}\). To simplify, we split the fraction into separate terms: \(\frac{64 x^{3}}{8 x^{3}} - \frac{72 x^{2}}{8 x^{3}} + \frac{12 x}{8 x^{3}}\).

For each term, we perform the following steps to simplify:
  • \(\frac{64 x^{3}}{8 x^{3}} \rightarrow 8 \) (by dividing 64 by 8 and canceling \({x^{3}}\))
  • \(\frac{72 x^{2}}{8 x^{3}} \rightarrow 9 x^{-1} \rightarrow \frac{9}{x}\) (by dividing 72 by 8 and balancing the exponents)
  • \(\frac{12 x}{8 x^{3}} \rightarrow \frac{3}{2} x^{-2} \rightarrow \frac{3}{2 x^{2}}\) (by simplifying the coefficients and adjusting the exponents accordingly)
Now the simplified result is \(8 - \frac{9}{x} + \frac{3}{2 x^{2}}\).
Polynomials
Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. Understanding polynomials is crucial to solving algebra problems.

The numerator in our exercise, \64x^{3} - 72x^{2} + 12x\, is a polynomial. It has three terms, each involving a different power of x and a coefficient:
  • \64x^{3}\ is the cubic term
  • \72x^{2}\ is the quadratic term
  • and \12x\ is the linear term.

When dividing this polynomial by \(8x^{3}\), each term is treated individually to simplify the algebraic expression further. This step-by-step approach helps in breaking down complex polynomials, making them more manageable and easier to understand.
Algebraic Expressions
Algebraic expressions combine numbers, variables, and arithmetic operations to represent a value. They form the foundation of algebra and are essential in solving equations and inequalities.

In the given problem, \(\frac{64x^{3}-72x^{2}+12x}{8x^{3}}\), we simplified an algebraic fraction by dividing each term of the polynomial in the numerator by the monomial in the denominator.

Each term of the expression was simplified separately:
  • The term \(\frac{64x^{3}}{8x^{3}} \rightarrow 8\)
  • The term \(\frac{72x^{2}}{8x^{3}} \rightarrow \frac{9}{x}\)
  • The term \(\frac{12x}{8x^{3}} \rightarrow \frac{3}{2x^{2}}\)

Simplifying each term individually and combining them results in a clearer and more understandable expression.
This skill is fundamental for students in learning how to manage and solve algebraic fractions effectively.

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Most popular questions from this chapter

Find each product. $$ (3 z+8)^{2} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(\frac{2 p}{q^{2}}\right)^{3}\left(\frac{3 p^{4}}{q^{-4}}\right)^{-1} $$

The factors in the following exercises involve fractions or decimals. Apply the methods of this section, and find each product. $$ \left(4 x-\frac{2}{3}\right)\left(4 x+\frac{2}{3}\right) $$

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ (g h)(-3) $$

The factors in the following exercises involve fractions or decimals. Apply the methods of this section, and find each product. $$ \left(5 r+\frac{2}{3} y\right)(r-5 y) $$
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