Problem 83 - Algebra and Trigonometry

Question

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ x^{3}-3 x^{2}-4 x+12=0 ;[-4,4] \text { by }[-15,15] $$

Step-by-Step Solution

Answer

Rational roots: \(x = -2\) and \(x = 2\).

Step 1: Understanding the Rational Root Theorem

The Rational Root Theorem states that any possible rational root of the polynomial equation \(x^3 - 3x^2 - 4x + 12 = 0\) will be a factor of the constant term (12) divided by a factor of the leading coefficient (1). In this case, the possible rational roots are factors of 12.

Step 2: Listing Possible Rational Roots

The factors of 12 are \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\). Since our leading coefficient is 1, the possible rational roots based on the Rational Root Theorem are these factors themselves: \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\).

Step 3: Graphing the Polynomial

By graphing the polynomial \(x^3 - 3x^2 - 4x + 12\) over the interval \([-4, 4]\) for \(x\) and \([-15, 15]\) for \(y\), we look to identify which of these potential roots cross the x-axis. This will tell us which of them are actual roots of the polynomial.

Step 4: Identifying Actual Rational Roots from the Graph

Upon graphing within the specified window, observe which intersections of the polynomial with the x-axis correspond to our list of possible rational roots. You should see exactly where the graph touches or crosses the x-axis within the interval. In this graph, the polynomial intersects the x-axis at \(x = -2, x = 2\).

Step 5: Conclusion on Rational Roots

Through graphing, it is confirmed that the real rational roots of the polynomial that fall within our viewing rectangle are \(x = -2\) and \(x = 2\). These values indeed satisfy the polynomial equation being set to zero.

Key Concepts

Polynomial Graphing

Graphing a polynomial is a crucial step in understanding its roots. A polynomial graph represents the visual behavior of the polynomial across different values of x. Using a graph, we can easily observe how the polynomial behaves. When we talk about graphing polynomials, the key points to look for are:

• Intercepts: Points where the graph crosses the x-axis (x-intercepts) and y-axis (y-intercepts).
• Turning Points: Peaks or valleys where the graph changes direction.
• End Behavior: The direction of the graph as x approaches infinity or negative infinity.

In our example, the polynomial is graphed over a specific interval, "[-4, 4]" for x, which helps us focus strictly on the area where real solutions are. By examining how the graph interacts with the x-axis, such as where it crosses or touches, we can identify its real roots. This interaction is indicative of solutions to the polynomial equation. The main advantage of graphing is the intuitive visual representation it provides, often revealing solutions that may not be obvious algebraically. But remember, always confirm these visually identified roots with algebraic checks for accuracy.

Real Solutions

Real solutions of a polynomial are the x-values where the polynomial equals zero. These are often referred to as roots or zeros of the polynomial. For our polynomial equation, we are interested in finding the real solutions, which, in the context of graphing, appear as the points where the curve hits the x-axis. Determining real solutions can involve:

• Graphing: Observing the graph to see where it crosses the x-axis.
• Factoring the Polynomial: Breaking down the polynomial into simpler expressions that are easier to solve.

In our example, the graph of the polynomial intersects the x-axis at two points, indicating two real solutions: at x = -2 and x = 2. These solutions satisfy the equation, meaning if you plug these x-values back into the polynomial, they make the equation zero. This confirms their validity as real solutions. Identifying real solutions is an important step in solving polynomial equations, as they can represent tangible scenarios or outcomes in real-world problems, underscoring their practical significance.

Factors of Polynomials

Understanding factors of polynomials helps in identifying possible rational roots. A polynomial's factors are expressions that multiply together to give the original polynomial.Factoring polynomials makes them easier to solve, especially when trying to identify their roots. For the polynomial \(x^3 - 3x^2 - 4x + 12\), using the Rational Root Theorem, we identify potential rational roots by focusing on the factors of the constant term.The process involves:

• Identifying Factors of the Constant Term: Here, the constant term is 12, with factors \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\).
• Pairing with Leading Coefficient's Factors: Since the leading coefficient is 1, the rational roots are these factors themselves.

This means we're examining all these potential rational roots by plugging them into the polynomial to see if they solve the equation. For instance, upon testing, both \(x = -2\) and \(x = 2\) satisfy the equation, confirming them as actual roots inside the specified viewing rectangle. Factoring showcases the polynomial's structure and is a practical technique in algebra for simplifying expressions and finding solutions.