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Chapter 1: Problem 17

Solve each equation by factoring. [Hint for Exer cises 19-22: First factor out a fractional power.] $$ 6 x^{5}=30 x^{4} $$

Short Answer

Expert verified
The solution to the equation is \(x = 5\).

Step by step solution

01

Identify Common Factors

Notice that both sides of the equation have a common factor. The terms on the left and right side have a common term, which is \(x^4\). Identify this as the greatest common factor (GCF).
02

Factor Out the GCF

Divide both terms by the common factor \(x^4\). This results in: \[ 6x^5 = 30x^4 \] Factor out \(x^4\) from both sides: \[ x^4(6x) = x^4(30) \]
03

Simplify the Equation

By dividing both sides by \(x^4\), we simplify the equation to: \[ 6x = 30 \]
04

Solve for \(x\)

Now, solve the equation for \(x\) by dividing both sides by 6: \[ x = \frac{30}{6} \] \[ x = 5 \]

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Key Concepts

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

Greatest Common Factor
The greatest common factor, or GCF, is an essential concept when factoring equations, especially in polynomial expressions. It represents the largest factor that divides two or more numbers or terms completely. In our example, we spotted that both sides of the equation, \(6x^5\) and \(30x^4\), share \(x^4\) as a common factor. Identifying the GCF allows us to simplify expressions by factoring it out.

To find the GCF in algebraic terms, you should:
  • Look for the highest power of each variable that appears in all terms.
  • Identify the smallest coefficient that divides all numerical coefficients.
This technique is crucial as it simplifies the equation, making it easier to solve. Once you factor out the GCF, you're often left with a simpler polynomial that reveals the solution more clearly.
Solving Polynomial Equations
Solving polynomial equations involves finding the value or values of the variable that make the equation true. Factorization is a powerful tool in this process. By expressing the equation in its simplest form, we can often reveal solutions that aren't immediately obvious.

Here's a basic approach to solving polynomial equations through factoring:
  • Identify any common factors, like we did with \(x^4\) in the exercise.
  • Factor out these common elements to simplify the equation.
  • Divide each term by these factors to further reduce the equation.
  • Once in a simpler form, solve the equation through basic algebraic operations like addition, subtraction, multiplication, or division.
In our provided exercise, this method turned a complex polynomial equation into a straightforward linear equation that was easily solved, illustrating the power of factoring in finding solutions quickly.
Fractional Powers in Algebra
Fractional powers in algebra are an extension of the concept of exponents. They represent roots of numbers or variables. For example, \(x^{1/2}\) is the same as \(\sqrt{x}\), where \(1/2\) signifies the square root. Understanding how to manage fractional powers is vital in factoring and simplifying polynomial equations.

Let's highlight some key points about fractional powers:
  • When you multiply powers with the same base, you add the exponents, including fractional ones.
  • Fractional exponents are another way to express roots, which helps in converting and solving equations.
  • Being comfortable with fractional powers broadens the range of equations you can solve.
In our equation, although not directly used, understanding fractional powers helps in recognizing other forms of exponents and facilitates deeper comprehension of algebraic manipulations needed when working with equations of varying complexity.

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Most popular questions from this chapter

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=\sqrt{x} $$

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ECONOMICS: Disposable Income Per capita disposable income (that is, after taxes have been subtracted) has fluctuated significantly in recent years, as shown in the following table. \begin{tabular}{lccccc} \hline Year & 2005 & 2006 & 2007 & 2008 & 2009 \\ \hline Percentage Change & \(0 \%\) & \(3.7 \%\) & \(5.6 \%\) & \(5.3 \%\) & \(2.7 \%\) \\\ \hline \end{tabular} a. Number the data columns with \(x\) -values \(1-5\) (so that \(x\) stands for years since 2004 ) and use quadratic regression to fit a parabola to the data. State the regression function. [Hint: See Example 10.] b. Use your regression curve to estimate the year and month when the percentage change was at its maximum.

The following function expresses an income tax that is \(10 \%\) for incomes below \(\$ 5000\), and otherwise is \(\$ 500\) plus \(30 \%\) of income in excess of \(\$ 5000\). \(f(x)=\left\\{\begin{array}{ll}0.10 x & \text { if } 0 \leq x<5000 \\\ 500+0.30(x-5000) & \text { if } x \geq 5000\end{array}\right.\) a. Calculate the tax on an income of \(\$ 3000\). b. Calculate the tax on an income of \(\$ 5000\). c. Calculate the tax on an income of \(\$ 10,000\). d. Graph the function.
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