Problem 45 - Algebra and Trigonometry

Question

Find all real solutions of the equation, correct to two decimals. $$ x^{3}-2 x^{2}-x-1=0 $$

Step-by-Step Solution

Answer

The real solutions are approximately 2.53, 0.33, and -1.86.

Step 1: Understand the Problem

The given problem is to find the real roots of the polynomial equation \( x^3 - 2x^2 - x - 1 = 0 \) with each solution rounded to two decimal places.

Step 2: Use the Rational Root Theorem

The Rational Root Theorem suggests that any rational root of the polynomial equation \( x^3 - 2x^2 - x - 1 = 0 \) is a factor of the constant term \(-1\). Possible rational roots are \( \pm 1 \).

Step 3: Test Possible Rational Roots

Substitute \( x = 1 \) and \( x = -1 \) into the equation:For \( x = 1 \):\( 1^3 - 2 \times 1^2 - 1 - 1 = -3 \), so \( x = 1 \) is not a root.For \( x = -1 \):\( (-1)^3 - 2 \times (-1)^2 - (-1) - 1 = -3 \), so \( x = -1 \) is not a root.Thus, there are no rational roots, indicating we need numerical methods, such as Newton's method or graphing, for real roots.

Step 4: Use Graphical or Numerical Method

Using numerical methods (such as graphing calculators or software), find the approximate roots by identifying where the graph of \( y = x^3 - 2x^2 - x - 1 \) crosses the x-axis.

Step 5: Approximate Roots

Using a graphing calculator or numerical software, observe that the polynomial crosses the x-axis around the points \( x \approx 2.53 \), \( x \approx 0.33 \), and \( x \approx -1.86 \). These are the approximate real roots of the polynomial, rounded to two decimals.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool when trying to find the rational roots of a polynomial equation. A "rational root" is a root that can be expressed as a fraction \( \frac{p}{q} \), where both \( p \) and \( q \) are integers. According to this theorem, if a polynomial has a rational root, then \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.

In our exercise, the polynomial is \( x^3 - 2x^2 - x - 1 = 0 \). Here, the constant term is \(-1\) and the leading coefficient is \(1\). This means possible rational roots are the factors of \(-1\), which are \( \pm 1 \).

• Test \( x = 1 \) and \( x = -1 \)
• Both substitutions result in non-zero values

This indicates that neither \( x = 1 \) nor \( x = -1 \) is a rational root. Thus, we conclude that the polynomial equation lacks rational roots, and other methods must be used to find the real roots.

Numerical Methods

When rational roots are absent, Numerical Methods come to the rescue. These methods help find approximate solutions for polynomial equations. One commonly used technique is Newton’s Method, but other software or algorithms may be used as well.

Numerical methods involve iteratively staying close to the actual root by performing calculations based on calculus and approximations. Using such techniques allows us to pinpoint the decimal values for roots with high precision. For the given polynomial, you can use a calculator or computer tool to approximate the roots where the polynomial expression equals zero.

• Set an initial guess
• Apply the iterative method to refine the approximation
• Repeat the process to achieve the desired precision

Through this, the solutions approximated for our polynomial were \( x \approx 2.53 \), \( x \approx 0.33 \), and \( x \approx -1.86 \), accurate to two decimal places.

Graphing Method

The Graphing Method is an intuitive way to approximate the roots of a polynomial equation. By plotting the graph of the polynomial function, you can visually identify where the curve crosses the x-axis, which indicates the real roots.

For our polynomial equation \( y = x^3 - 2x^2 - x - 1 \), a graphing calculator or graphing software can be used.

• Plot the polynomial function on a suitable range
• Observe the points of intersection with the x-axis
• Use a zoom tool to get a clearer view of the intersection points

By using this method, the graph reveals that the polynomial crosses the x-axis near \( x \approx 2.53 \), \( x \approx 0.33 \), and \( x \approx -1.86 \). Checking these approximate values against the graph helps confirm the accuracy of the roots found using numerical methods. Graphing is quite useful for visual learners, providing a clear idea of where the roots lie on the x-axis.