Problem 40 - Algebra and Trigonometry

Question

\(31-42\) . Solve the equation both algebraically and graphically. $$ 2 x^{5}-243=0 $$

Step-by-Step Solution

Answer

The solution is approximately \(x = 2.72\).

Step 1: Arrange equation properly

Start with the given equation: \(2x^5 - 243 = 0\). The equation is already set to zero, which is a good starting point for solving algebraically.

Step 2: Isolate the polynomial term

Add 243 to both sides to isolate the polynomial term: \(2x^5 = 243\).

Step 3: Solve for x^5

Divide both sides by 2 to solve for \(x^5\): \(x^5 = \frac{243}{2}\).

Step 4: Find the root

Take the fifth root of both sides to find \(x\): \(x = \sqrt[5]{rac{243}{2}}\). Calculate this to get an approximate value: \(x \approx 2.72\).

Step 5: Graph the equation

To solve graphically, plot the function \(y = 2x^5 - 243\). The x-axis represents where the function equals zero.

Step 6: Identify x-intercept

Look for the x-intercept of the graph, which is where the curve crosses the x-axis. This point represents the solution, confirming that \(x \approx 2.72\).

Key Concepts

Algebraic Solution

Solving polynomial equations like \(2x^5 - 243 = 0\) involves finding the value(s) of \(x\) that make the equation true. Start by isolating the term with the polynomial degree. For \(2x^5 - 243 = 0\), add 243 to both sides to have \(2x^5 = 243\). The next step is to simplify this to \(x^5\) by dividing both sides by 2, yielding \(x^5 = \frac{243}{2}\).

• Isolating terms is crucial for simplification.
• Each operation is aimed at gradually simplifying the equation to reach \(x\).

The algebraic solution process continues by taking the fifth root of each side. This finds the value of \(x\) since \(x\) raised to the fifth power would approximately equal \(\frac{243}{2}\). Hence, \(x = \sqrt[5]{\frac{243}{2}}\) leads to an approximate result of \(x \approx 2.72\). This step, known as finding the root, provides a numerical solution to the polynomial equation.

• Taking roots is a reverse of the exponentiation process.
• The fifth root specifically is used because \(x\) is raised to the power of five.

Graphical Solution

Graphing the equation provides a visual method for finding solutions to polynomial equations such as \(2x^5 - 243 = 0\). By plotting the function \(y = 2x^5 - 243\), you can observe where it crosses the x-axis (where \(y = 0\)).

• Plotting starts by determining a range of \(x\) values and computing \(y\).
• The x-axis crossing points reflect where the function equals zero.

Seeing the curve intersect the x-axis at \(x \approx 2.72\) corroborates the algebraic findings. This x-intercept signifies the solution point or root of the function. Graphing can be useful to confirm algebraic results and provide a deeper understanding of the function's behavior.

• Graphical solutions aid in visualizing how polynomial functions behave.
• The solution visually confirms what was derived algebraically.

Fifth Root

The fifth root is essential in solving polynomial equations where the variable is raised to the fifth power, such as \(x^5 = \frac{243}{2}\). Extracting the fifth root is akin to finding a number which, when raised to the power of five, results in \(\frac{243}{2}\).

• Taking the fifth root is solving for \(x\) in a simpler form.
• The root indicates balancing the equation resulting from the power of five.

In our problem, the fifth root finds \(x\) and shows that the approximate solution is \(x \approx 2.72\). This demonstrates the power of roots in simplifying and solving high-degree polynomial equations efficiently. This concept is pivotal because it allows breaking down more complex polynomial expressions into comprehensible terms.

• Fifth roots can make solving non-linear equations manageable.
• It reflects the inverse operation of raising to the fifth power.