Problem 17 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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).

Step 2: 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: \[ x^4(6x) = x^4(30) \]

Step 3: Simplify the Equation

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

Step 4: Solve for \(x\)

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

Key Concepts

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.