Polynomial Technology

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Answer Key & Explanations

Polynomial Technology · Grade 12 · Worksheet 1

1. A biomedical engineer is modeling the concentration of a new drug in a patient's bloodstream using the polynomial function C(t) = -0.02t³ + 0.3t² + 0.5t, where t represents hours after administration and C(t) is measured in milligrams per liter. The drug is considered effective when the concentration is increasing and safe when the concentration is decreasing. Determine the time interval during which the drug is both effective and safe according to this model. Answer: From t = 0 to t = 5 hours Solution: In pharmaceutical modeling, the effectiveness and safety of drugs often depend on whether their concentration is increasing or decreasing in the bloodstream.
   Full step-by-step solution

   In pharmaceutical modeling, the effectiveness and safety of drugs often depend on whether their concentration is increasing or decreasing in the bloodstream. This can be determined by analyzing the derivative of the concentration function. When the derivative is positive, the concentration is rising; when negative, it's falling. For polynomial functions, finding critical points where the derivative equals zero helps identify these transition points between increasing and decreasing behavior.

2. Use Desmos to graph f(x)=x³-9x²+23x-15 and find the sum of all x-coordinates where the local maximum and minimum occur. Answer: 6 Solution: Find the derivative of f(x)=x³-9x²+23x-15 f'(x)=3x²-18x+23 Set the derivative equal to zero to find critical points 3x²-18x+23=0 x=[18±√(324-276)]/6 x=[18±√48]/6 x=[18±4√3]/6 x=3±(2√3)/3 The two critical points are x=3+2√3/3 and x=3-2√3/3 (3+2√3/3)+(3-2√3/3)=6 The answer is 6.
   Full step-by-step solution

   Step 1: Find the derivative of f(x)=x³-9x²+23x-15 f'(x)=3x²-18x+23 Step 2: Set the derivative equal to zero to find critical points 3x²-18x+23=0 Step 3: Use the quadratic formula to solve x=[18±√(324-276)]/6 x=[18±√48]/6 x=[18±4√3]/6 x=3±(2√3)/3 Step 4: The two critical points are x=3+2√3/3 and x=3-2√3/3 Step 5: Sum the x-coordinates of the local extrema (3+2√3/3)+(3-2√3/3)=6 The answer is 6.

3. A biomedical engineer is modeling the concentration of a new drug in a patient's bloodstream using the polynomial function C(t) = -0.02t³ + 0.3t² + 0.5t, where C is the concentration in milligrams per liter and t is time in hours after administration. The drug is considered effective when the concentration is increasing and above 1 mg/L. Determine the time interval during which the drug remains effective. Answer: (2, 5) Solution: In pharmaceutical modeling, we often analyze when drugs reach therapeutic levels.
   Full step-by-step solution

   In pharmaceutical modeling, we often analyze when drugs reach therapeutic levels. The increasing concentration condition relates to the derivative being positive, while the minimum concentration requirement involves solving an inequality. This type of analysis helps determine optimal dosing schedules in medical applications.

4. Liam is analyzing the profit function for his tech startup, which is modeled by P(x) = -2x³ + 15x² - 24x + 10, where x represents thousands of units sold and P(x) is profit in thousands of dollars. He wants to determine the intervals where his profit is increasing and decreasing to optimize production. Find the critical points and determine on which intervals the profit function is increasing. Answer: Increasing on (1, 4) Solution: P(x) = -2x³ + 15x² - 24x + 10 P'(x) = -6x² + 30x - 24 Set derivative equal to zero to find critical points -6x² + 30x - 24 = 0 Divide the entire equation by -6: x² - 5x + 4 = 0 (x - 1)(x - 4) = 0 x = 1 and x = 4 The critical points split the x-axis into three intervals: (-∞, 1), (1, 4), (4, ∞)…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Find the derivative of P(x)** The profit function is P(x) = -2x³ + 15x² - 24x + 10 Differentiate: P'(x) = -6x² + 30x - 24 --- **Step 2: Set derivative equal to zero to find critical points** -6x² + 30x - 24 = 0 Divide the entire equation by -6: x² - 5x + 4 = 0 Factor: (x - 1)(x - 4) = 0 So critical points are: x = 1 and x = 4 --- **Step 3: Determine intervals** The critical points split the x-axis into three intervals: (-∞, 1), (1, 4), (4, ∞) --- **Step 4: Test the sign of P'(x) in each interval** We have P'(x) = -6x² + 30x - 24, or equivalently -6(x - 1)(x - 4). Since -6 is negative, the sign of P'(x) is opposite to the sign of (x - 1)(x - 4). Check each interval: - For x in (-∞, 1): pick x = 0 (x - 1) = negative, (x - 4) = negative Product (x - 1)(x - 4) = positive P'(x) = -6 × positive = negative → decreasing - For x in (1, 4): pick x = 2 (x - 1) = positive, (x - 4) = negative Product (x - 1)(x - 4) = negative P'(x) = -6 × negative = positive → increasing - For x in (4, ∞): pick x = 5 (x - 1) = positive, (x - 4) = positive Product (x - 1)(x - 4) = positive P'(x) = -6 × positive = negative → decreasing --- **Step 5: Conclusion** The profit function is increasing on the interval (1, 4). --- **Final answer:** Increasing on (1, 4)

5. A polynomial function f(x) is graphed on a coordinate plane. The graph shows the function crossing the x-axis at (-2, 0), (1, 0), and (3, 0). It touches but does not cross the x-axis at (0, 0). The graph passes through the point (2, -4). Determine the equation of the polynomial in factored form, assuming it has the lowest possible degree with real coefficients. Answer: f(x) = -x^2(x + 2)(x - 1)(x - 3) Solution: The roots of a polynomial correspond to the x-intercepts of its graph. When a graph crosses the axis, the root has odd multiplicity; when it touches without crossing, the root has even multiplicity. The lowest degree polynomial will use the smallest multiplicities that match this behavior…
   Full step-by-step solution

   The roots of a polynomial correspond to the x-intercepts of its graph. When a graph crosses the axis, the root has odd multiplicity; when it touches without crossing, the root has even multiplicity. The lowest degree polynomial will use the smallest multiplicities that match this behavior (typically 1 for crossing, 2 for touching). The leading coefficient can be found by substituting a known point into the factored form and solving for the constant.

6. A polynomial function f(x) is graphed on a coordinate plane. The graph shows the function crossing the x-axis at (-3, 0), (1, 0), and (4, 0), and it touches but does not cross the x-axis at (2, 0). The graph passes through the point (0, -24). Determine the equation of the polynomial in factored form, assuming it has the smallest possible degree and a leading coefficient of 1. Answer: f(x) = (x + 3)(x - 1)(x - 4)(x - 2)^2 Solution: The x-intercepts of a polynomial graph correspond to the real roots of the function. When a graph crosses the x-axis at an intercept, the corresponding factor has an odd multiplicity. When the graph touches but does not cross the x-axis, the corresponding factor has an even multiplicity (at…
   Full step-by-step solution

   The x-intercepts of a polynomial graph correspond to the real roots of the function. When a graph crosses the x-axis at an intercept, the corresponding factor has an odd multiplicity. When the graph touches but does not cross the x-axis, the corresponding factor has an even multiplicity (at least 2). The y-intercept provides a specific point that the equation must satisfy, which can be used to confirm the correct structure of the factored form.