Polynomial Technology

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Polynomial Technology · Grade 12 · Worksheet 2

1. A biomedical engineer is modeling the concentration of a new medication in a patient's bloodstream over time. The concentration C(t) in milligrams per liter is given by the polynomial function C(t) = -0.02t³ + 0.45t² - 2.7t + 5 for 0 ≤ t ≤ 15 hours. At what time does the medication reach its maximum concentration in the bloodstream, and what is that maximum concentration? Answer: t = 9 hours, C = 8.42 mg/L Solution: In optimization problems, finding maximum or minimum values of a function involves analyzing its derivative.
   Full step-by-step solution

   In optimization problems, finding maximum or minimum values of a function involves analyzing its derivative. The derivative represents the rate of change, and when this rate changes from positive to negative, the function reaches a maximum point. This concept applies to various real-world situations like maximizing profit, minimizing cost, or optimizing drug dosage in medical applications.

2. A polynomial function f(x) is graphed on a coordinate plane. The graph shows three x-intercepts at (-2, 0), (1, 0), and (3, 0). The graph passes through the point (0, -6) and has end behavior where f(x) → -∞ as x → -∞ and f(x) → +∞ as x → +∞. Determine the equation of the polynomial function in factored form. Answer: f(x) = (x + 2)(x - 1)(x - 3) Solution: When analyzing polynomial graphs, the x-intercepts correspond to the roots of the function, which become the factors in the factored form.
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   When analyzing polynomial graphs, the x-intercepts correspond to the roots of the function, which become the factors in the factored form. The end behavior reveals information about the degree and leading coefficient - if both ends go in opposite directions, the degree is odd, and the specific direction indicates the sign of the leading coefficient. The y-intercept provides additional information to determine any scaling factor.

3. Use Desmos to graph f(x) = x^4 - 18x^3 + 109x^2 - 252x + 180 and find the sum of all distinct real zeros. Answer: 18 Solution: Graph f(x) = x^4 - 18x^3 + 109x^2 - 252x + 180 using Desmos or a graphing calculator. Identify the x-intercepts (real zeros). From the graph, the zeros are at x = 3, x = 4, x = 5, and x = 6.
   Full step-by-step solution

   Step 1: Graph f(x) = x^4 - 18x^3 + 109x^2 - 252x + 180 using Desmos or a graphing calculator. Step 2: Identify the x-intercepts (real zeros). From the graph, the zeros are at x = 3, x = 4, x = 5, and x = 6. Step 3: Since all zeros are distinct, sum them: 3 + 4 + 5 + 6 = 18. The answer is 18.

4. Use Desmos to graph f(x)=x³-9x²+23x-15 and find the sum of all x-coordinates where the graph has local extrema. 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 Use technology (Desmos or graphing calculator) to solve the quadratic equation.
   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 technology (Desmos or graphing calculator) to solve the quadratic equation. Using the quadratic formula: x = [18 ± sqrt(324 - 276)] / 6 x = [18 ± sqrt(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.

5. Liam is designing a roller coaster and needs to analyze the polynomial function f(x) = 2x³ - 9x² - 24x + 5, which models the track's elevation in meters at position x kilometers from the start. He needs to determine the coordinates of all local extrema points to ensure safety regulations are met. Find the x-coordinates where the roller coaster reaches its highest and lowest points between positions. Answer: x = -1 and x = 4 Solution: To find the local extrema of the function f(x) = 2x³ - 9x² - 24x + 5, we need to find the critical points. Critical points occur where the first derivative is zero or undefined.
   Full step-by-step solution

   To find the local extrema of the function f(x) = 2x³ - 9x² - 24x + 5, we need to find the critical points. Critical points occur where the first derivative is zero or undefined. Since this is a polynomial, the derivative is defined everywhere, so we set the first derivative equal to zero. Step 1: Find the first derivative, f'(x). The derivative of 2x³ is 6x². The derivative of -9x² is -18x. The derivative of -24x is -24. The derivative of the constant 5 is 0. So, f'(x) = 6x² - 18x - 24. Step 2: Set the first derivative equal to zero and solve for x. 6x² - 18x - 24 = 0 Step 3: Simplify the equation by dividing every term by the greatest common factor, which is 6. (6x²)/6 - (18x)/6 - (24)/6 = 0/6 This simplifies to: x² - 3x - 4 = 0 Step 4: Solve the quadratic equation x² - 3x - 4 = 0. We can factor this equation. We look for two numbers that multiply to -4 and add to -3. These numbers are -4 and +1. So, the factored form is (x - 4)(x + 1) = 0. Step 5: Set each factor equal to zero and solve for x. x - 4 = 0 gives x = 4. x + 1 = 0 gives x = -1. These two x-values, x = -1 and x = 4, are the critical points where the slope of the function is zero. This means the roller coaster track is horizontal at these positions, indicating they are the locations of potential local maximum (highest point) or local minimum (lowest point) elevations. Therefore, the x-coordinates where the roller coaster reaches its highest and lowest points are x = -1 and x = 4. (Note: To confirm which is the maximum and which is the minimum, you would use the second derivative test. The second derivative is f''(x) = 12x - 18. For x = -1, f''(-1) = 12(-1) - 18 = -30, which is negative, indicating a local maximum. For x = 4, f''(4) = 12(4) - 18 = 30, which is positive, indicating a local minimum.)

6. Use Desmos to graph f(x)=x³-9x²+26x-21 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+26x-21. f'(x)=3x²-18x+26. Set the derivative equal to zero to find critical points: 3x²-18x+26=0.
   Full step-by-step solution

   Step 1: Find the derivative of f(x)=x³-9x²+26x-21. f'(x)=3x²-18x+26. Step 2: Set the derivative equal to zero to find critical points: 3x²-18x+26=0. Step 3: Use the quadratic formula to solve: x = [18 ± sqrt(324 - 312)] / 6 = [18 ± sqrt(12)] / 6 = [18 ± 2√3] / 6 = 3 ± √3/3. Step 4: The two critical points are x = 3 + √3/3 and x = 3 - √3/3. Step 5: Sum the x-coordinates: (3 + √3/3) + (3 - √3/3) = 6. The answer is 6.

7. Use Desmos to graph f(x) = x⁴ - 10x³ + 35x² - 50x + 25 and find the sum of the y-coordinates of all local minima. Answer: 0 Solution: Graph f(x) = x⁴ - 10x³ + 35x² - 50x + 25 using Desmos or a graphing calculator. Observe the shape. The function is a quartic (degree 4) with a positive leading coefficient, so it opens upward.
   Full step-by-step solution

   Step 1: Graph f(x) = x⁴ - 10x³ + 35x² - 50x + 25 using Desmos or a graphing calculator. Step 2: Observe the shape. The function is a quartic (degree 4) with a positive leading coefficient, so it opens upward. It has two local minima and one local maximum. Step 3: Use the graph to identify the local minima. The first local minimum occurs near x = 1.382, with y ≈ 0. The second local minimum occurs near x = 3.618, with y ≈ 0. Step 4: Sum the y-coordinates: 0 + 0 = 0. The answer is 0.

8. A polynomial function f(x) is graphed on a coordinate plane. The graph shows the function crossing the x-axis at (-2, 0) and (1, 0), and touching the x-axis at (3, 0). The graph passes through the point (0, -6). If the function has a positive leading coefficient, what is the equation of the polynomial in factored form? Answer: f(x) = (x + 2)(x - 1)(x - 3)^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 a root, that root has odd multiplicity (typically 1).
   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 a root, that root has odd multiplicity (typically 1). When the graph touches but does not cross the x-axis, the root has even multiplicity (typically 2). The general factored form includes these factors raised to their respective multiplicities. A point on the graph can be used to solve for any leading coefficient if needed.