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Chapter 1: Problem 13

Solve each equation by factoring. [Hint for Exer cises 19-22: First factor out a fractional power.] $$ 5 x^{3}-20 x=0 $$

Short Answer

Expert verified
The solutions are \(x = 0\), \(x = 2\), and \(x = -2\).

Step by step solution

01

Identify the Common Factor

Examine the equation \(5x^3 - 20x = 0\) to identify any common factors in all terms. Both terms contain the factor \(5x\).
02

Factor Out the Greatest Common Factor (GCF)

Factor \(5x\) out from each term in the equation: \[5x(x^2 - 4) = 0\]
03

Recognize Special Products

Notice that \(x^2 - 4\) is a difference of squares, which can be factored further using the identity \(a^2 - b^2 = (a-b)(a+b)\).
04

Factor the Difference of Squares

Apply the difference of squares formula to \(x^2 - 4\):\[(x-2)(x+2)\]So the equation becomes:\[5x(x-2)(x+2) = 0\]
05

Solve Each Factor for Zero

Set each factor equal to zero and solve for \(x\):1. \(5x = 0\) implies \(x = 0\)2. \(x - 2 = 0\) implies \(x = 2\)3. \(x + 2 = 0\) implies \(x = -2\)

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Key Concepts

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

Understanding the Greatest Common Factor
In algebra, identifying the Greatest Common Factor (GCF) is a crucial first step in factoring expressions. The GCF is the largest factor that divides two or more terms in a polynomial evenly. To find the GCF in an expression like \(5x^3 - 20x\), you look for the most significant factor that is common in all terms.
- The coefficients of the terms are 5 and 20. The greatest common divisor of these numbers is 5.
- Both terms also share the variable \(x\), and since both terms are divisible by at least one \(x\), \(x\) is part of the GCF.

Therefore, the GCF of the expression \(5x^3 - 20x\) is \(5x\). Factoring the GCF simplifies the expression to \(5x(x^2 - 4)\). This makes it easier to see other factoring opportunities, such as recognizing a difference of squares.
Exploring the Difference of Squares
Recognizing and working with the difference of squares is a fundamental algebraic skill. The difference of squares refers to an expression in the form \(a^2 - b^2\), which can be factored into \((a-b)(a+b)\).
In the equation \(5x(x^2 - 4) = 0\), the expression \(x^2 - 4\) fits the difference of squares pattern:
- Here \(x^2\) is \(a^2\) and \(4\) is \(2^2\).
- Applying the difference of squares formula, \(x^2 - 4\) becomes \((x-2)(x+2)\).

This transformation is powerful because it breaks down complex polynomials into simpler binomial factors, which are easier to handle when solving equations.
Mastering Solving Quadratic Equations via Factoring
Once a quadratic equation is factored completely, finding the solutions is straightforward. Consider the factored equation resulting from the original exercise: \(5x(x-2)(x+2) = 0\). Each factor is set to zero to find the possible solutions for \(x\). This step is based on the Zero Product Property, which states that if the product of multiple factors is zero, at least one of them must be zero.
- For \(5x = 0\), solve by dividing both sides by 5, giving \(x = 0\).
- For \(x - 2 = 0\), solve by adding 2 to both sides, resulting in \(x = 2\).
- For \(x + 2 = 0\), solve by subtracting 2 from both sides, yielding \(x = -2\).

Thus, the solutions of the quadratic equation are \(x = 0, 2,\) and \(-2\). These steps systematically break down the problem and allow you to solve quadratic equations efficiently by factoring.

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Most popular questions from this chapter

BEHAVIORAL SCIENCES: Smoking and Educatior According to a study, the probability that a smoker will quit smoking increases with the smoker's educational level. The probability (expressed as a percent) that a smoker with \(x\) years of education will quit is approximately \(y=0.831 x^{2}-18.1 x+137.3\) (for \(10 \leq x \leq 16\) ). a. Graph this curve on the window \([10,16]\) by \([0,100]\). b. Find the probability that a high school graduate smoker \((x=12)\) will quit. c. Find the probability that a college graduate smoker \((x=16)\) will quit.

GENERAL: Newsletters A newsletter has a maximum audience of 100 subscribers. The publisher estimates that she will lose 1 reader for each dollar she charges. Therefore, if she charges \(x\) dollars, her readership will be \((100-x)\). a. Multiply this readership by \(x\) (the price) to find her total revenue. Multiply out the resulting quadratic function. b. What price should she charge to maximize her revenue? [Hint: Find the value of \(x\) that maximizes this quadratic function.]

How will the graph of \(y=-(x-4)^{2}+8\) differ from the graph of \(y=-x^{2} ?\) Check by graphing both functions together.

63-64. BUSINESS: Straight-Line Depreciation Straight-line depreciation is a method for estimating the value of an asset (such as a piece of machinery) as it loses value ( \({ }^{\prime \prime}\) depreciates" \()\) through use. Given the original price of an asset, its useful lifetime, and its scrap value (its value at the end of its useful lifetime), the value of the asset after \(t\) years is given by the formula: $$ \begin{aligned} \text { Value }=(\text { Price })-&\left(\frac{(\text { Price })-(\text { Scrap value })}{(\text { Useful lifetime })}\right) \cdot t \\ & \text { for } 0 \leq t \leq(\text { Useful lifetime }) \end{aligned} $$ a. A farmer buys a harvester for $$\$ 50,000$$ and estimates its useful life to be 20 years, after which its scrap value will be $$\$ 6000$$. Use the formula above to find a formula for the value \(V\) of the harvester after \(t\) years, for \(0 \leq t \leq 20\). b. Use your formula to find the value of the harvester after 5 years. c. Graph the function found in part (a) on a graphing calculator on the window \([0,20]\) by \([0,50,000]\). [Hint: Use \(x\) instead of \(t\).]

SOCIAL SCIENCE: Immigration The percentage of immigrants in the United States has changed since World War I as shown in the following table. \begin{tabular}{llllllllll} \hline Decade & 1930 & 1940 & 1950 & 1960 & 1970 & 1980 & 1990 & 2000 & 2010 \\\ \hline Immigrant Percentage & \(11.7\) & \(8.9\) & 7 & \(5.6\) & \(4.9\) & \(6.2\) & \(7.9\) & \(11.3\) & \(12.5\) \\ \hline \end{tabular} a. Number the data columns with \(x\) -values \(0-8\) (so that \(x\) stands for decades since 1930 ) and use 0 quadratic regression to fit a parabola to the data. State the regression function. [Hint: See Example 10.] b. Use your curve to estimate the percentage in \(2016 .\)
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