Polynomial Technology

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Polynomial Technology · Grade 12 · Worksheet 1

1. Sophia is an environmental engineer modeling the water flow rate in a river after a storm. The flow rate F(t) in thousands of cubic meters per hour is given by the polynomial function F(t) = -0.02t³ + 0.48t² - 2.88t + 6.4, where t is the time in hours after the storm begins (t ≥ 0). Environmental regulations require that for safety, the flow rate must be decreasing and must remain below 3.2 thousand cubic meters per hour. Using a graphing calculator or Desmos, determine the time interval(s) during which the river flow rate meets both safety conditions. Answer: (4, 8) Solution: Find the derivative of F(t). F'(t) = -0.06t² + 0.96t - 2.88. Set F'(t) = 0 to find critical points.
   Full step-by-step solution

   Step 1: Find the derivative of F(t). F'(t) = -0.06t² + 0.96t - 2.88. Step 2: Set F'(t) = 0 to find critical points. -0.06t² + 0.96t - 2.88 = 0. Multiply by -100: 6t² - 96t + 288 = 0. Divide by 6: t² - 16t + 48 = 0. Factor: (t - 4)(t - 12) = 0. So t = 4 and t = 12. Step 3: Test intervals around critical points using F'(t). For t < 4, test t = 0: F'(0) = -2.88 < 0, so F is decreasing. For 4 < t < 12, test t = 6: F'(6) = -0.06(36) + 0.96(6) - 2.88 = -2.16 + 5.76 - 2.88 = 0.72 > 0, so F is increasing. For t > 12, test t = 14: F'(14) = -0.06(196) + 0.96(14) - 2.88 = -11.76 + 13.44 - 2.88 = -1.2 < 0, so F is decreasing. Step 4: F is decreasing on (0, 4) and (12, ∞). Step 5: Use technology to graph F(t) and the line y = 3.2. Find where F(t) < 3.2. Using Desmos or a graphing calculator, the graph shows F(t) = 3.2 at t = 4 and t = 8. For 0 < t < 4, F(t) > 3.2. For 4 < t < 8, F(t) < 3.2. For t > 8, F(t) > 3.2 again. Step 6: Combine conditions: F decreasing AND F < 3.2. On (0, 4), F is decreasing but F > 3.2. On (12, ∞), F is decreasing but F > 3.2 (after t = 8). The only interval where both conditions hold is (4, 8). Final answer: (4, 8).

2. Kaia uses a graphing calculator to analyze the polynomial f(x) = -3x^5 + 15x^3 - 12x. She observes the graph and notes the x-intercepts and the behavior around them. Using the calculator, determine all real zeros of the polynomial and classify each zero as having odd or even multiplicity based on whether the graph crosses or touches the x-axis at that point. Then, write the polynomial in completely factored form over the real numbers. Answer: Zeros: x = -2 (odd, crosses), x = -1 (odd, crosses), x = 0 (odd, crosses), x = 1 (odd, crosses), x = 2 (odd, crosses); Factored form: f(x) = -3x(x - 2)(x + 2)(x - 1)(x + 1) Solution: Factor out the greatest common factor. The polynomial is f(x) = -3x^5 + 15x^3 - 12x. All terms have a factor of -3x, so factor that out: f(x) = -3x(x^4 - 5x^2 + 4).
   Full step-by-step solution

   Step 1: Factor out the greatest common factor. The polynomial is f(x) = -3x^5 + 15x^3 - 12x. All terms have a factor of -3x, so factor that out: f(x) = -3x(x^4 - 5x^2 + 4). Step 2: Factor the quartic expression. Let u = x^2, then x^4 - 5x^2 + 4 = u^2 - 5u + 4. Factor this quadratic: u^2 - 5u + 4 = (u - 4)(u - 1). Substitute back: x^4 - 5x^2 + 4 = (x^2 - 4)(x^2 - 1). Step 3: Factor further using difference of squares. x^2 - 4 = (x - 2)(x + 2), and x^2 - 1 = (x - 1)(x + 1). Step 4: Write the completely factored form: f(x) = -3x(x - 2)(x + 2)(x - 1)(x + 1). Step 5: Identify the zeros. Set each factor equal to zero: x = 0, x = 2, x = -2, x = 1, x = -1. All zeros are distinct (each factor has exponent 1), so all have odd multiplicity. Step 6: Verify using a graphing calculator. Input f(x) = -3x^5 + 15x^3 - 12x into Desmos or a graphing calculator. The graph shows the function crossing the x-axis at x = -2, x = -1, x = 0, x = 1, and x = 2, confirming all zeros have odd multiplicity (since the graph crosses, not touches). Final answer: The real zeros are x = -2, x = -1, x = 0, x = 1, and x = 2, all with odd multiplicity (crosses). The factored form is f(x) = -3x(x - 2)(x + 2)(x - 1)(x + 1).

3. A biomedical engineer is modeling the concentration of a new medication in a patient's bloodstream using the polynomial function C(t) = -0.02t³ + 0.36t² - 1.8t + 2.4, where C(t) represents the concentration in milligrams per liter and t is the time in hours after administration. The medication becomes effective when the concentration reaches its maximum value. At what time does the medication become effective, and what is the maximum concentration? Answer: t = 6 hours, C = 2.4 mg/L Solution: In optimization problems involving polynomial functions, we use calculus to find extreme values.
   Full step-by-step solution

   In optimization problems involving polynomial functions, we use calculus to find extreme values. The derivative of a function gives us its rate of change, and setting this derivative equal to zero helps identify critical points where the function may reach maximum or minimum values. For real-world applications like drug concentration modeling, we need to verify which critical point represents the maximum by checking the behavior of the function or using the second derivative test.

4. 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.

5. Consider the polynomial function f(x) = x^5 - 5x^4 + 5x^3 + 5x^2 - 6x. Using technology to analyze the graph, determine how many distinct real roots this function has. Answer: 5 Solution: Step 1: The function is f(x) = x^5 - 5x^4 + 5x^3 + 5x^2 - 6x Step 2: Factor out x: f(x) = x(x^4 - 5x^3 + 5x^2 + 5x - 6) Step 3: Using technology to graph the function, we can see it crosses the x-axis at x = -1, x = 0, x = 1, x = 2, and x = 3 Step 4: Verify these are roots: f(-1) = 0, f(0) = 0,…
   Full step-by-step solution

   Step 1: The function is f(x) = x^5 - 5x^4 + 5x^3 + 5x^2 - 6x Step 2: Factor out x: f(x) = x(x^4 - 5x^3 + 5x^2 + 5x - 6) Step 3: Using technology to graph the function, we can see it crosses the x-axis at x = -1, x = 0, x = 1, x = 2, and x = 3 Step 4: Verify these are roots: f(-1) = 0, f(0) = 0, f(1) = 0, f(2) = 0, f(3) = 0 Step 5: Since these are 5 distinct x-values where f(x) = 0, the function has 5 distinct real roots The answer is 5.

6. Use Desmos to graph f(x) = x^4 - 10x^3 + 35x^2 - 50x + 24 and find the sum of all distinct real zeros. Answer: 10 Solution: Graph f(x) = x^4 - 10x^3 + 35x^2 - 50x + 24 using Desmos or a graphing calculator. Observe the x-intercepts (zeros) of the graph. The graph crosses the x-axis at x = 1, x = 2, x = 3, and x = 4.
   Full step-by-step solution

   Step 1: Graph f(x) = x^4 - 10x^3 + 35x^2 - 50x + 24 using Desmos or a graphing calculator. Step 2: Observe the x-intercepts (zeros) of the graph. The graph crosses the x-axis at x = 1, x = 2, x = 3, and x = 4. Step 3: Verify these zeros by factoring: f(x) = (x-1)(x-2)(x-3)(x-4). Step 4: The distinct real zeros are 1, 2, 3, and 4. Step 5: Sum the zeros: 1 + 2 + 3 + 4 = 10. The answer is 10.

7. A polynomial function f(x) is graphed on a coordinate plane. The graph crosses the x-axis at (-4, 0) and (2, 0), touches and turns at (-1, 0), and passes through the point (1, 6). The end behavior shows that as x → -∞, f(x) → -∞ and as x → ∞, f(x) → ∞. What is the equation of this polynomial in standard form? Answer: f(x) = x^3 + 3x^2 - 6x - 8 Solution: Identify the roots and their multiplicities from the graph description. - The graph crosses at x = -4 and x = 2, so these are roots with odd multiplicity (multiplicity 1).
   Full step-by-step solution

   Step 1: Identify the roots and their multiplicities from the graph description. - The graph crosses at x = -4 and x = 2, so these are roots with odd multiplicity (multiplicity 1). - The graph touches and turns at x = -1, so this is a root with even multiplicity (multiplicity 2). Step 2: Write the polynomial in factored form. f(x) = a(x + 4)(x - 2)(x + 1)^2 Step 3: Use the point (1, 6) to find the value of a. Substitute x = 1 and f(x) = 6 into the equation: 6 = a(1 + 4)(1 - 2)(1 + 1)^2 6 = a(5)(-1)(2)^2 6 = a(5)(-1)(4) 6 = a(-20) a = 6/(-20) = -3/10 Step 4: Write the polynomial with the determined a. f(x) = (-3/10)(x + 4)(x - 2)(x + 1)^2 Step 5: Expand the polynomial to standard form. First, expand (x + 4)(x - 2) = x^2 + 2x - 8 Then, expand (x^2 + 2x - 8)(x + 1)^2 = (x^2 + 2x - 8)(x^2 + 2x + 1) = x^2(x^2 + 2x + 1) + 2x(x^2 + 2x + 1) - 8(x^2 + 2x + 1) = x^4 + 2x^3 + x^2 + 2x^3 + 4x^2 + 2x - 8x^2 - 16x - 8 = x^4 + 4x^3 - 3x^2 - 14x - 8 Step 6: Multiply by the coefficient a = -3/10. f(x) = (-3/10)(x^4 + 4x^3 - 3x^2 - 14x - 8) = -3/10x^4 - 12/10x^3 + 9/10x^2 + 42/10x + 24/10 = -3/10x^4 - 6/5x^3 + 9/10x^2 + 21/5x + 12/5 Step 7: Check the end behavior. As x → -∞, f(x) → -∞ (matches given) As x → ∞, f(x) → ∞ (matches given) The answer is f(x) = -3/10x^4 - 6/5x^3 + 9/10x^2 + 21/5x + 12/5.