Problem 44 - College Algebra with Modeling and Visualization

Question

Solve the polynomial inequality (a) symbolically and (b) graphically. $$ 2 x^{3} \leq 3 x^{2}+5 x $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -1) \cup (0, 2.5]\).

Step 1: Move all terms to one side

Start by rewriting the inequality with all terms on one side. Subtract \(3x^2 + 5x\) from both sides to get: \[2x^3 - 3x^2 - 5x \leq 0\]

Step 2: Factor the polynomial

Factor the terms in the expression \(2x^3 - 3x^2 - 5x\).First, factor out the common factor \(x\):\[x(2x^2 - 3x - 5) \leq 0\]Next, factor the quadratic \(2x^2 - 3x - 5\) using the quadratic formula, considering the roots.

Step 3: Find the roots of the quadratic equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) to find the roots of \(2x^2 - 3x - 5\). Here, \(a = 2\), \(b = -3\), and \(c = -5\):\[x = \frac{3 \pm \sqrt{(-3)^2-4 \cdot 2 \cdot (-5)}}{2 \cdot 2}\]Calculate the discriminant \((-3)^2-4 \cdot 2 \cdot (-5) = 9 + 40 = 49\), which is a perfect square:\[x = \frac{3 \pm 7}{4}\]This gives roots \(x = 2.5\) and \(x = -1\).

Step 4: List all critical points

The critical points of the inequality come from the factor \(x = 0\) and the roots \(x = 2.5\), and \(x = -1\) found in the previous step. These points are \(-1, 0,\) and \(2.5\).

Step 5: Test the intervals

Test the sign of the polynomial \(x(2x^2 - 3x - 5)\) in each interval divided by the critical points: - Interval \((-\infty, -1)\): Pick \(x = -2\), substitute into the factored form: \[-2(2(-2)^2 - 3(-2) - 5) = -2(8 + 6 - 5) = -2 \times 9 = -18 < 0\] - Interval \((-1, 0)\): Pick \(x = -0.5\), substitute: \[-0.5(2(-0.5)^2 - 3(-0.5) - 5) = -0.5(0.5 + 1.5 - 5) = -0.5 \times -3 = 1.5 > 0\]- Interval \((0, 2.5)\): Pick \(x = 1\), substitute: \[1(2 \cdot 1^2 - 3 \cdot 1 - 5) = 1(-6) = -6 < 0\]- Interval \((2.5, \infty)\): Pick \(x = 3\), substitute: \[3(2 \cdot 3^2 - 3 \cdot 3 - 5) = 3(18 - 9 - 5) = 3 \times 4 = 12 > 0\]From this, the inequality is satisfied for \(x \in (-\infty, -1) \cup (0, 2.5]\).

Step 6: Graphically examining the solution

Graph the polynomial \(2x^3 - 3x^2 - 5x\) and identify where it is below or equal to the x-axis. The graph intersects the x-axis at \(x = -1, 0,\) and \(2.5\). The polynomial is below or touches the x-axis in the same intervals found numerically: \((-\infty, -1)\) and \((0, 2.5]\).

Step 7: Write the final solution

Combine the results to describe where the inequality holds true both symbolically and graphically. The solution to the inequality \(2x^3 \leq 3x^2 + 5x\) is when \(x \in (-\infty, -1) \cup (0, 2.5]\).

Key Concepts

Quadratic Formula

The quadratic formula is a universal method used to find the roots of a quadratic equation, which is of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here's how it works in practice:

• Calculate the discriminant, \( b^2 - 4ac \). This value helps determine the nature of the roots:
• If it's positive, you have two distinct real roots.
• If it's zero, you have one real root, or a repeated root.
• If it's negative, the roots are complex.
• Substitute the values of \( a \), \( b \), and \( c \) into the formula to find the roots.

In the example given, the quadratic \( 2x^2 - 3x - 5 \) has coefficients \( a=2 \), \( b=-3 \), \( c=-5 \). Substituting these values gives roots at \( x = 2.5 \) and \( x = -1 \), with a discriminant of 49, confirming two distinct real roots.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components (factors) that, when multiplied back together, will give the original polynomial. This method is useful in solving polynomial inequalities or equations because it allows us to find the roots or "solutions" easily. To factor a polynomial like \( 2x^3 - 3x^2 - 5x \), here’s what to do:

• Check for a common factor among all the terms. In this case, \( x \) is common, so we factor it out: \[ x(2x^2 - 3x - 5) \]
• To further factor, deal with the quadratic part \( 2x^2 - 3x - 5 \). This step often requires using the quadratic formula, especially when the quadratic doesn’t factor into integers easily.

After applying the quadratic formula, the quadratic factors as \( (x - 2.5)(x + 1) \). Therefore, the entire polynomial can be represented as \( x(x - 2.5)(x + 1) \). This factored form helps to easily identify the critical points and analyze the behavior of the inequality.

Graphical Analysis

Graphical analysis involves visualizing the behavior of a polynomial on a graph to understand where the polynomial is greater than, less than, or equal to zero. For a cubic equation like \( 2x^3 - 3x^2 - 5x \), it can touch or intersect the x-axis at several points, which signify the roots or solutions. To conduct this analysis:

• Graph the polynomial. Use the roots obtained from factoring as the x-intercepts.
• On the graph, observe where the curve is below or above the x-axis. In polynomial inequalities, you often identify intervals where the curve lies below the x-axis to satisfy inequalities like \( \, \leq 0 \).

In our example, after graphing, we note that the curve is below the x-axis on the intervals \( (-\infty, -1) \) and \( (0, 2.5] \). These are critical for symbolic solution confirmation.

Critical Points

Critical points are the x-values where a function changes its sign, crosses the x-axis or where the polynomial is equal to zero. These points help segment the real number line into regions where the polynomial is either positive, zero, or negative. Critical points are vital because they provide the thresholds for testing intervals over which you determine the sign of your polynomial expression. For the inequality \( 2x^3 \leq 3x^2 + 5x \), the critical points were \( -1, 0, \) and \( 2.5 \):

• These values are found by setting each factor in the factored form \( x(x - 2.5)(x + 1) \) to zero.
• Once you have the critical points, divide the number line into intervals around these points.
• Test the polynomial’s sign within each interval to determine which intervals satisfy the inequality.

In the described solution, testing around these values revealed the intervals \( (-\infty, -1) \) and \( (0, 2.5] \) as sections that satisfy the original inequality \( 2x^3 \leq 3x^2 + 5x \).