Finding Roots Of Polynomial Equation X^4 + X^2 = 4x^3 - 12x + 12

In this comprehensive guide, we will explore how to determine the roots of the polynomial equation x⁴ + x² = 4x³ - 12x + 12. We will employ a graphing calculator and a system of equations to identify both integer and non-integer roots, rounding the latter to the nearest hundredth. This approach combines visual analysis with algebraic techniques, offering a robust method for solving polynomial equations.

Understanding Polynomial Roots

The roots of a polynomial equation are the values of x that make the equation equal to zero. These roots are also known as solutions or zeros of the polynomial. Finding the roots of a polynomial is a fundamental problem in algebra, with applications in various fields such as engineering, physics, and economics. Polynomial equations can have real roots, which are numbers that can be plotted on a number line, and complex roots, which involve imaginary numbers. For this particular equation, we will focus on finding the real roots.

Rearranging the Equation

Before we can use a graphing calculator or a system of equations, it is essential to rewrite the given equation in standard polynomial form. This involves moving all terms to one side of the equation, setting it equal to zero. Starting with the original equation:

x⁴ + x² = 4x³ - 12x + 12

Subtract 4x³, add 12x, and subtract 12 from both sides to get:

x⁴ - 4x³ + x² + 12x - 12 = 0

Now, we have the polynomial equation in the standard form, which is much easier to work with for both graphing and algebraic methods.

Using a Graphing Calculator to Find Roots

A graphing calculator is a powerful tool for visualizing polynomial functions and approximating their roots. The roots correspond to the points where the graph of the function intersects the x-axis. Here’s how to use a graphing calculator to find the roots of our equation:

1. Input the Equation

First, enter the polynomial function y = x⁴ - 4x³ + x² + 12x - 12 into the graphing calculator. This is typically done in the “Y=” menu.

2. Adjust the Viewing Window

Next, adjust the viewing window to get a clear picture of the graph. The standard window might not show all the important features of the polynomial, such as all the x-intercepts. You may need to experiment with the window settings to find a suitable view. A good starting point is often to set the x-axis range from -5 to 5 and the y-axis range from -20 to 20. This can be adjusted based on the behavior of the graph.

3. Identify the X-Intercepts

The x-intercepts are the points where the graph crosses the x-axis. These points represent the real roots of the equation. From the graph, you should be able to see the approximate locations of the roots. Our polynomial appears to have roots near x = -1.73, x = 2 and x = 1.73.

4. Use the Calculator’s Root-Finding Feature

Most graphing calculators have a built-in function to find roots (or zeros) of a function. This feature typically requires you to specify a left bound and a right bound for the interval in which the root lies, as well as an initial guess. The calculator then uses numerical methods to find a more accurate approximation of the root. Using this feature, we find roots close to -1.73, 2, and 2.73. This requires more precise calculation, but the graphing calculator makes the process straightforward.

5. Approximate Non-Integer Roots

For non-integer roots, the calculator will provide decimal approximations. Round these approximations to the nearest hundredth as required. In our case, we will round the roots to two decimal places, giving us -1.73 and 2.73.

By using the graphing calculator, we have visually identified the roots and obtained accurate approximations for the non-integer roots. This method provides a quick and intuitive way to solve polynomial equations.

Solving with a System of Equations

Another approach to finding the roots of the polynomial equation involves using a system of equations. This method can be particularly useful when you suspect that the polynomial can be factored. Factoring a polynomial involves expressing it as a product of simpler polynomials, which can then be solved individually.

1. Look for Rational Root Possibilities

The Rational Root Theorem states that if a polynomial has integer coefficients, then any rational root of the polynomial must be of the form ±(factor of the constant term) / (factor of the leading coefficient). In our equation, x⁴ - 4x³ + x² + 12x - 12 = 0, the constant term is -12 and the leading coefficient is 1. Thus, the possible rational roots are ±1, ±2, ±3, ±4, ±6, and ±12.

2. Test Possible Roots Using Synthetic Division

Synthetic division is a method for dividing a polynomial by a linear factor of the form (x - c), where c is a constant. If the remainder after synthetic division is zero, then c is a root of the polynomial. Let’s test the possible rational roots we identified in the previous step.

For example, let's test x = 2:

2 | 1 -4 1 12 -12
 | 2 -4 -6 12
 ---------------------
 1 -2 -3 6 0

Since the remainder is 0, x = 2 is a root of the polynomial. The quotient from the synthetic division is x³ - 2x² - 3x + 6, which represents the remaining polynomial after dividing out the factor (x - 2).

3. Factor the Reduced Polynomial

Now we need to find the roots of the reduced polynomial x³ - 2x² - 3x + 6 = 0. We can try factoring by grouping:

x²(x - 2) - 3(x - 2) = 0

(x² - 3)(x - 2) = 0

4. Solve for the Remaining Roots

We now have the polynomial factored as (x - 2)(x² - 3) = 0. Setting each factor equal to zero gives us:

• x - 2 = 0 => x = 2
• x² - 3 = 0 => x² = 3 => x = ±√3

5. Approximate Non-Integer Roots

The roots we found are x = 2, x = √3, and x = -√3. Approximating the non-integer roots to the nearest hundredth, we get:

• √3 ≈ 1.73
• -√3 ≈ -1.73

Thus, the roots of the polynomial equation are approximately -1.73, 1.73, and 2. This method combines algebraic factoring techniques with the Rational Root Theorem to find the exact and approximate roots of the polynomial.

Comparing the Results

Both the graphing calculator method and the system of equations (factoring) method have yielded the same roots for the polynomial equation x⁴ - 4x³ + x² + 12x - 12 = 0. The roots are approximately -1.73, 1.73, and 2.

The graphing calculator provides a visual and numerical approximation, allowing for quick identification of roots, especially non-integer ones. The system of equations method, particularly factoring, allows for finding exact roots when possible and can be a more algebraic approach.

Analyzing the Given Options

Now, let's examine the given options to determine which one correctly identifies the roots of the polynomial equation:

A. -12, 20 B. -2.73, 2, 2.73 C. -1.73, 1.73, 2 D. -20, 12

Comparing the roots we found (-1.73, 1.73, and 2) with the options provided, we can see that option C matches our results. Therefore, the correct answer is:

C. -1.73, 1.73, 2

Conclusion

In summary, we have successfully found the roots of the polynomial equation x⁴ + x² = 4x³ - 12x + 12 using both a graphing calculator and a system of equations. By rearranging the equation, using the graphing calculator to visualize the roots, and applying factoring techniques, we determined that the roots are approximately -1.73, 1.73, and 2. This exercise demonstrates the power of combining graphical and algebraic methods to solve polynomial equations, providing a comprehensive approach to root finding. Understanding these techniques is essential for solving complex mathematical problems and applying them in various real-world scenarios.