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Chapter 1: Problem 35

Perform the multiplication or division and simplify. $$\frac{\frac{x^{3}}{x+1}}{\frac{x}{x^{2}+2 x+1}}$$

Short Answer

Expert verified
The simplified expression is \( x^{3} + x^{2} \).

Step by step solution

01

Identify the Division

We are given the expression \( \frac{\frac{x^{3}}{x+1}}{\frac{x}{x^{2}+2x+1}} \). This is a division of two fractions. We can rewrite this division of fractions as a multiplication by flipping the second fraction.
02

Rewrite as Multiplication

The expression becomes \( \frac{x^{3}}{x+1} \times \frac{x^{2} + 2x + 1}{x} \). This is because dividing by a fraction is the same as multiplying by its reciprocal.
03

Simplify the Expression

Notice that \( x^{2} + 2x + 1 \) is a perfect square and can be factored into \( (x+1)^2 \). Thus, the expression becomes \( \frac{x^{3}}{x+1} \times \frac{(x+1)^{2}}{x} \).
04

Cancel Common Factors

We can cancel one factor of \( x+1 \) in the numerator with the \( x+1 \) in the denominator. This simplifies the expression to \( \frac{x^{3}(x+1)}{x} \).
05

Simplify Further

Now, perform the multiplication to get \( \frac{x^{4} + x^{3}}{x} \). Finally, separate the fraction to get \( \frac{x^{4}}{x} + \frac{x^{3}}{x} \).
06

Final Simplification

Simplify each term: \( \frac{x^{4}}{x} = x^{3} \) and \( \frac{x^{3}}{x} = x^{2} \). Put them together to get \( x^{3} + x^{2} \).

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Key Concepts

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

Multiplication of Fractions
When dealing with the multiplication of fractions, an important thing to remember is that you multiply the numerators and the denominators across. Let鈥檚 clarify this with an example. If you have two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), the multiplication would be \( \frac{a \times c}{b \times d} \).
In the case of algebraic fractions, the method is the same. Let's say we have \( \frac{x^3}{x+1} \times \frac{(x+1)^2}{x} \). Here:
  • The numerators are \( x^3 \) and \( (x+1)^2 \), so you multiply these together.
  • The denominators are \( x+1 \) and \( x \), which are also multiplied.
This results in: \( \frac{x^3 \times (x+1)^2}{x+1 \times x} \).
Multiplication of fractions becomes straightforward when you understand this pattern and allows you to handle more complex algebraic fractions with ease.
Simplifying Expressions
Simplifying expressions means making them as concise and straightforward as possible. The goal is to reduce fractions or algebraic terms to their simplest forms. For example, when given the expression \( \frac{x^3 \times (x+1)^2}{x+1 \times x} \), simplifying it involves canceling out common factors sol that it has the simplest form possible.
The expression involves a common factor \( x+1 \) in the numerator and denominator. By canceling one \( x+1 \) from both, we are left with:
  • Numerator: \( x^3 \times (x+1) \)
  • Denominator: \( x \)
Thus, the expression simplifies to \( \frac{x^3(x+1)}{x} \).
This step is crucial in order not to lose sight of the identity of an expression while simplifying it.
Factoring Polynomials
Factoring polynomials is a process of breaking down a complex polynomial into simpler, multiplied factors. Recognizing patterns aids significantly in this task. In our expression, we observed \( x^2 + 2x + 1 \). This expression is a perfect square trinomial that factors as \( (x+1)^2 \).
This recognition helps in simplifying other parts of an exercise or problem. Here are some tips to keep in mind:
  • Identify special patterns like perfect squares: \( a^2 + 2ab + b^2 = (a+b)^2 \).
  • Look for a common factor that can be factored out from the terms.
  • Always double-check by redistributing to ensure the factors are correct.
The ability to factor polynomials provides the toolkit to simplify complex algebraic expressions and solve equations easily. It鈥檚 a basic yet powerful skill in algebra.

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Most popular questions from this chapter

Distances in a City \(A\) city has streets that run north and south and avenues that run east and west, all equally spaced. Streets and avenues are numbered sequentially, as shown in the figure. The walking distance between points \(A\) and \(B\) is 7 blocks - that is, 3 blocks east and 4 blocks north. To find the straight-line distance \(d\), we must use the Distance Formula. (a) Find the straight-line distance (in blocks) between \(A\) and \(B\) (b) Find the walking distance and the straight-line distance between the corner of 4 th St. and 2 nd Ave. and the corner of 11 th St. and 26 th Ave. (c) What must be true about the points \(P\) and \(Q\) if the walking distance between \(P\) and \(Q\) equals the straight[line distance between \(P\) and \(Q ?\)

If you had a million \(\left(10^{6}\right)\) dollars in a suitcase, and you spent a thousand (10 ) dollars each day, how many years would it take you to use all the money? Spending at the same rate, how many years would it take you to empty a suitcase filled with a billion ( \(10^{9}\) ) dollars?

Radicals Simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers. (a) \(\sqrt[5]{x^{3} y^{2}} \sqrt[19]{x^{4} y^{16}}\) (b) \(\frac{\sqrt[3]{8 x^{2}}}{\sqrt{x}}\)

As of July 2013 , the population of the United States was \(3.164 \times 10^{8},\) and the national debt was \(1.674 \times 10^{13}\) dollars. How much was each person's share of the debt?

Area of a Region Sketch the region in the coordinate plane that satisfies both the inequalities \(x^{2}+y^{2} \leq 9\) and \(y \geq|x| .\) What is the area of this region?
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