Problem 24 - Applied Calculus

Question

Solve each equation using a graphing calculator. Round answers to two decimal places. $$ x^{6}+2 x^{5}-5 x^{4}=0 $$

Step-by-Step Solution

Answer

The roots are 0 (multiplicity 4), 1.45, and -3.45.

Step 1: Factor the Equation

First, we identify a common factor in the equation. Notice that each term has a factor of \(x^4\). Factoring out \(x^4\), we have:\[ x^4(x^2 + 2x - 5) = 0 \]This reveals two possibilities: \(x^4 = 0\) or \(x^2 + 2x - 5 = 0\).

Step 2: Solve for x from x^4 = 0

For the equation \(x^4 = 0\), taking the fourth root of both sides, we find:\[ x = 0 \]This is a repeated root with multiplicity 4.

Step 3: Solve the Quadratic Equation x^2 + 2x - 5 = 0

Next, solve the quadratic equation \(x^2 + 2x - 5 = 0\) using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, \(a = 1\), \(b = 2\), and \(c = -5\). Plug these values into the formula:\[ x = \frac{-2 \pm \sqrt{2^2 - 4\cdot1\cdot(-5)}}{2\cdot1} \]\[ x = \frac{-2 \pm \sqrt{4 + 20}}{2} \]\[ x = \frac{-2 \pm \sqrt{24}}{2} \]\[ x = \frac{-2 \pm 2\sqrt{6}}{2} \]\[ x = -1 \pm \sqrt{6} \]

Step 4: Approximate the Roots

Using a calculator, approximate the values for the expression \(-1 \pm \sqrt{6}\):\[ x_1 = -1 + \sqrt{6} \approx 1.45 \]\[ x_2 = -1 - \sqrt{6} \approx -3.45 \]

Step 5: Combine All Roots

Combine all roots from the solutions:1. The root from \(x^4 = 0\) is \(x = 0\) with a multiplicity of 4.2. The roots from \(x^2 + 2x - 5 = 0\) are approximately \(x = 1.45\) and \(x = -3.45\).The solutions are: \(x = 0\), \(x \approx 1.45\), and \(x \approx -3.45\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a key method used to simplify complex equations. It involves breaking down a polynomial into simpler 'factors' that, when multiplied together, give the original polynomial.
For the equation given, each term has a common factor of \(x^4\). By factoring \(x^4\) out of the equation \(x^6 + 2x^5 - 5x^4 = 0\), we get:

• \(x^4(x^2 + 2x - 5) = 0\)

This simplification not only reveals that \(x = 0\) is a root but also leaves a quadratic equation \(x^2 + 2x - 5 = 0\) to solve. Faktoring is essential as it converts a higher degree polynomial into a product of simpler polynomials, making it easier to solve equations.

Quadratic Formula

The quadratic formula is a powerful tool for solving equations of the form \(ax^2 + bx + c = 0\). It allows us to find the roots of any quadratic equation easily through the expression:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

For the quadratic equation \(x^2 + 2x - 5 = 0\), we identify:

• \(a = 1\)
• \(b = 2\)
• \(c = -5\)

Plugging these values into the formula,

• \(x = \frac{-2 \pm \sqrt{24}}{2}\)

Using this formula, the exact roots \(x = -1 \pm \sqrt{6}\) are found. The quadratic formula is invaluable when factoring is difficult or impractical, providing a straightforward mechanical procedure to arrive at the solution.

Root Multiplicity

Root multiplicity refers to the number of times a particular root appears as a solution of a polynomial equation. When we have a root with higher multiplicity, it essentially means that the root is repeated that many times.
In our equation, \(x = 0\) appears because of the \(x^4\) factor. This means \(x = 0\) is a root with multiplicity 4, as it would satisfy the equation by being repeated four times in the solution of \(x^4 = 0\). Identifying multiplicities is important in understanding the behavior of polynomial functions, especially in their graphs at intersections and turning points.

Approximating Roots with a Calculator

Sometimes, roots involve irrational numbers, making it challenging to provide their exact value. In these cases, calculators become handy tools for approximating these roots to a desired decimal place.
For the equation provided, solving analytically gives us roots \(-1 \pm \sqrt{6}\), which are difficult to express exactly without a calculator. By using a graphing calculator or a scientific calculator, the approximate values:

• \(-1 + \sqrt{6} \approx 1.45\)
• \(-1 - \sqrt{6} \approx -3.45\)

When equations are factored into forms that include irrational numbers or complex expressions, approximating roots ensures a practical and understandable result is achieved. Calculators efficiently provide these approximations, which are useful in both academics and real-world applications.