Problem 23 - Basic Technical Mathematics with Calculus

Question

Use a calculator to solve the given equations to the nearest 0.01. $$8 x^{4}+36 x^{3}+35 x^{2}-4 x-4=0$$

Step-by-Step Solution

Answer

Approximate roots: -3.55, -2.31, 0.60, 1.01 (rounded to nearest 0.01).

Step 1: Understanding the Equation

The equation given is a quartic polynomial, which is a polynomial of degree 4: \( 8x^4 + 36x^3 + 35x^2 - 4x - 4 = 0 \). Our task is to find the roots of this equation using a calculator. Since this is a nontrivial polynomial, exact solutions might not be easily found, thus the requirement to use a calculator to find approximate solutions to the nearest hundredth.

Step 2: Input the Equation into the Calculator

To solve this equation using a calculator, input the polynomial \( 8x^4 + 36x^3 + 35x^2 - 4x - 4 \) into a polynomial solver function. Many scientific calculators with polynomial solving capability or graphing calculators will have this feature. Enter all coefficients in the correct order into the solver.

Step 3: Interpret the Results

Once the equation is input correctly, use the calculator to compute the roots. The calculator will output approximate values for these roots. Typically, there may be up to four roots for a quartic equation, depending on their multiplicity.

Step 4: Round the Results

The calculator will provide numerical approximations with several decimal places. Round these results to the nearest 0.01, as required by the problem statement. Ensure each root is checked and rounded properly.

Key Concepts

Quartic Polynomial

A quartic polynomial is a type of polynomial where the highest exponent of the variable is four. In simpler words, it is a polynomial of degree four. For example, the given equation in this exercise, \( 8x^4 + 36x^3 + 35x^2 - 4x - 4 = 0 \), is a quartic polynomial because the term with the highest power in the expression is \( x^4 \).
Quartic polynomials can have:

• Up to 4 real or complex roots.
• Terms involving \( x^0 \) (constant term) through \( x^4 \).

Understanding the structure of these polynomials helps us know how many solutions or roots there might be. When solving them, we typically find the values of \( x \) that make the polynomial equal to zero. These roots could be real numbers or complex numbers. However, identifying these roots exactly becomes complex with higher-degree polynomials, such as quartics, which is why numerical methods or calculators are useful for finding approximate solutions.

Root Approximation

When dealing with higher-degree polynomials like quartic polynomials, finding the exact roots can be quite challenging without advanced techniques. Root approximation becomes a very practical approach, particularly when we need solutions close to certain real values. This process involves estimating roots to a specific degree of accuracy. Calculators or numerical algorithms approximate the roots because:

• Exact algebraic solutions might be impossible, especially for nontrivial polynomials.
• Calculations can be highly complex and time-consuming if done manually.

Therefore, we often rely on calculators to provide numerical roots, which we then round to a required precision, like the nearest hundredth. This means if the calculator outputs a root as 1.23456, we round it to 1.23 to adhere to the exercise's requirement of the nearest two decimal places.

Calculator Usage

Utilizing a calculator to find approximate roots of a polynomial can greatly streamline the problem-solving process. Here's how you can effectively use a calculator to solve a quartic polynomial:

Steps for Using a Calculator

Input the complete polynomial equation into the calculator’s polynomial solver. Make sure the calculator used has this specific functionality, which is common in many scientific and graphing calculators.

• Navigate to the polynomial solver mode if your calculator has one.
• Input the coefficients in precise order from the highest degree term to the lowest, i.e., 8, 36, 35, -4, -4 for this example.
• Confirm that the equation is entered correctly to avoid any errors.
• Let the calculator solve and generate approximate root values.

Once you get the results, interpret the values by rounding them to the nearest hundredth as per the requirement. For instance, if a root obtained is 3.456, round it to 3.46. This method is reliable for handling complex problems swiftly while ensuring sufficient accuracy for most practical purposes.