Problem 17 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The solution is \(x = 5\).

Step 1: Rewrite the Equation

Begin by rewriting the given equation. The original equation is:\[6x^{5} = 30x^{4}\].

Step 2: Factor Out Common Terms

Identify the common terms on both sides of the equation. Here, both terms are divisible by \(6x^4\). Factor this out:\[6x^{4}(x) = 6x^{4}(5)\].

Step 3: Simplify the Equation

Divide both sides by \(6x^4\) to isolate \(x\):\[x = 5\].

Step 4: Verify the Solution

Substitute \(x = 5\) back into the original equation to check if both sides are equal:\[6(5)^5 = 30(5)^4\].Calculate both sides to confirm:\[6 imes 3125 = 30 imes 625\].Indeed, both sides equal \(18750\), confirming our solution is correct.

Key Concepts

Factoring Polynomials

Factoring polynomials is a technique used to solve equations where polynomials, which are expressions consisting of variables and coefficients, are present. When you are asked to solve an equation by factoring, you are trying to rewrite the equation in a way that easier solutions can be derived.

To factor a polynomial, first look for common factors in each term. In our example, the equation is \(6x^{5} = 30x^{4}\). Both terms share the common factor \(6x^4\).

• By factoring out \(6x^4\), we simplify the equation to \(6x^{4}(x) = 6x^{4}(5)\)
• This shows that each side has a common component: \(6x^4\)

Reducing equations in this manner makes it possible to solve for the unknown variable by focusing only on the remaining factors. Factoring polynomials is especially useful for more complex algebraic equations, turning them into simpler, solvable forms.

Fractional Powers

Fractional powers involve exponents that are fractions, such as in expressions where the variable is raised to a power that is a fraction. However, in the context of our problem, we're more concerned about spotting powers of variable expressions that can be easily manipulated.

In our exercise, the term \(x^5\) is more cumbersome than \(x\). By factoring, we transform the difficult terms into something easier to manage. Here, the fractional sense comes when stripping down the larger powers, which helps in reducing the complexity of calculations. For example, recognizing that \(x^{5}\) can be rewritten as \(x^{4} imes x\), allows us to see the term more clearly in fractions or smaller powers to factor out redundant components.

• In the given equation \(6x^5=30x^4\), breaking down the powers brings clarity
• You factor out \(x^4\) to focus on what remains: namely, \(x\)

This mental step helps frame more complex expressions in a simpler mathematical form by effectively "reducing" powers through fractional understanding.

Verify Solutions

Once you've found a potential solution to an equation, verifying your result is crucial. This ensures the solution makes sense within the context of the original problem. Verification involves substituting the solution back into the original equation to check if it holds true.

For our example \(x = 5\), substituting 5 back into the initial equation \(6x^5 = 30x^4\), translates to: \(6(5)^5 = 30(5)^4\). If both sides are equal when evaluated:

• Calculate \(6 \times 3125 = 18750\)
• Calculate \(30 \times 625 = 18750\)

This equality assures us that the solution \(x = 5\) is indeed correct. Verifying solutions prevents errors and confirms the reliability of your factored pathways. Always perform this crucial step to solidify your confidence in the solutions you derive.