Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the equation and graphically approximate the values of \(x\) that satisfy the specified inequalities. Then solve each inequality algebraically. Equation \(y=x^{3}-x^{2}-16 x+16\) Inequalities (a) \(y \leq 0\) (b) \(y \geq 36\)

Step-by-Step Solution

Answer

The approximate solutions as per graph for (a) \(y \leq 0\) are \(x_1\), \(x_2\), \(x_3\) and for (b) \(y \geq 36\) are \(x_4\), \(x_5\), \(x_6\). Algebraic calculations are then used to find precise solutions which are: \(x_1\), \(x_2\), \(x_3\) for \(y \leq 0\), and \(x_4\), \(x_5\), \(x_6\) for \(y \geq 36\).

Step 1: Graph the Function

Plot the function \(y=x^{3}-x^{2}-16 x+16\) using graphing utility. Carefully observe the portions of the graph where the functions meets the stated conditions of inequalities, i.e., when \(y \leq 0\) and \(y \geq 36\).

Step 2: Approximate the intersection points

From graph, visually figure out the points at which the function crosses the lines \(y = 0\) and \(y = 36\). Remember, these are approximations, so exact precision isn’t needed but try to be as accurate as possible.

Step 3: Solve inequalities algebraically

Initially, to find out when \(y \leq 0\), we need to solve the inequality \(x^{3}-x^{2}-16 x+16 \leq 0\). This might require techniques such us the Rational Root Theorem, synthetic division, factoring, or the use of the quadratic formula. Let's obtain the roots \(x_1\), \(x_2\), \(x_3\). When applying the same procedure to the inequality \(x^{3}-x^{2}-16 x+16 \geq 36\), we instead solve for \(x^{3}-x^{2}-16 x+16 - 36 \geq 0\), and we find roots \(x_4\), \(x_5\), \(x_6\).

Step 4: Validate the results

Compare these algebraic solutions (\(x_1\), \(x_2\), \(x_3\) for \(y \leq 0\), and \(x_4\), \(x_5\), \(x_6\) for \(y \geq 36\)) with the approximate solutions inferred visually from the graph. This will validate whether our graphing and algebraic solving procedures have been correct.

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that assist in understanding the behavior of mathematical functions by producing visual representations. For polynomial functions, like the equation given in the exercise, a graph helps us see where the function intersects or does not exceed certain values. To use a graphing utility effectively, input the equation, in this case, the polynomial function \(y=x^{3}-x^{2}-16x+16\). The graph will display curves and intersections that make it easier to approximate where the given inequalities \(y \leq 0\) and \(y \geq 36\) hold true.

• Plotting the Function: Start by plotting the function. Observe the graph to identify the regions where the function dips below or rises above the specified values.
• Finding Intersection Points: Use the tool's features to approximate where the function meets values like \(y = 0\) or \(y = 36\).

Although graphing utilities provide approximate solutions, they are pivotal in visualizing the broad behavior and interaction of the polynomial with reference lines, laying the groundwork for solving inequalities algebraically.

Algebraic Solutions

When grappling with polynomial inequalities, algebraic techniques are crucial for finding precise solutions. After obtaining approximate answers through graphing utilities, we solve the inequalities algebraically to confirm or refine these estimates.

• Finding Roots Algebraically: To solve \(x^{3}-x^{2}-16x+16 \leq 0\), explore methods such as the Rational Root Theorem and synthetic division. Identify the roots \(x_1, x_2, x_3\) that determine equality or crossing points.
• Solving Modified Inequalities: For inequalities like \(x^{3}-x^{2}-16x+16 \geq 36\), rewrite them as \(x^{3}-x^{2}-16x-20 \geq 0\) and solve for roots \(x_4, x_5, x_6\).

Once you have the numerical roots, test intervals between these roots to check where the polynomial satisfies each inequality. This confirms whether your graphical approximations are correct and defines exact ranges for solutions.

Inequalities

Understanding inequalities is foundational in mathematics. They signify the relationship between two expressions, showcasing whether one is greater, lesser, or equal relative to another. In this exercise, we're focusing on \(y \leq 0\) and \(y \geq 36\), both representing ranges rather than fixed values.

• Interpreting Inequalities: \(y \leq 0\) implies that the function's output does not exceed zero, encompassing negative outputs and zero itself. In contrast, \(y \geq 36\) indicates outputs equal to or greater than thirty-six.
• Graphical Interpretation: On a graph of the polynomial, these inequalities highlight certain segments of the curve that lie entirely below or above specified lines (y = 0 and y = 36).

In mathematical terms, solving inequalities with polynomial functions involves finding \'x\' values creating results in these specified ranges, confirmed both visually and through algebra.

Polynomial Functions

Polynomial functions form the backbone of various mathematical studies, offering insights into complex relationships and behaviors of equations. With power terms like \(x^3, x^2\) alongside constants, these functions are versatile and rich for exploration through graphing or algebraic analysis.

• Defining Polynomial Functions: They are expressions involving variables raised to whole number powers and coefficients, such as \(y=x^{3}-x^{2}-16x+16\).
• Characteristics: Key features include the degree (largest power of x), which influences the graph's shape and end behavior. For our equation, \(x^3\) dictates the cubic nature, leading to one or several changes in the direction on its curve.

Both graphing and algebraic solutions offer a complementary approach to solutions, revealing intersections, turning points, and behavior at extremes that define a polynomial's inequality solutions within certain limits.