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LApreC2010-106matrixaprtakehome_4r

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Life After preCalculus 2010
Session 106
matrix

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LApreC2010-106matrixaprtakehome_4r

  1. 1. Math4R April Take Home Exam Name:_______________ Part I: Polynomial Zeros BackgroundLet’s solve Quadratic Equations with Complex coefficients using theQuadratic Formula!Theorem 1.1Let f(x) = x2 + α x + β where α , β ∈ C. If f(x) = 0, then x = ∈ C.Lemma 1.1.1If f(x2) = x4 + α x2 + β where α , β ∈ C, then x = ∈ C.Lemma 1.1.2If f(x3) = x6 + α x3 + β where α , β ∈ C, then x = ∈ C.Scipione del Ferro (1465 – 1526) discovered the following Cubic Formula.Theorem 1.2One root of the depressed cubic equation x3 = α x + β where α , β ∈ ℜ is: x=+Rafael Bombelli (1526 – 1572) made the following related assertion.Accordingly, the root given by del Ferro must be Real!Theorem 1.3The sum of the cube roots of complex conjugates is in fact a Real number.This number, in turn, is the sum of two complex conjugates. + = (a2 + b2i) + (a2 – b2i) = 2a2 + 0i ∈ ℜThen Girolamo Cardano (1501 – 1576) stated the following.Theorem 1.4If f(x) = x3 + α 1x2 + β 1x + γ 1 = 0, then g(y) = f(y – = y3 + α 2y +β 2 = 0.del Ferro’s Theorem amounts to a Cubic Formula to solve Depressed CubicEquations. Further, Cardan’s Theorem allows us to “depress” any CubicEquation. Now we can solve any Cubic Equation! How does all this work?Well, its really the Quadratic Formula in disguise since Cardano also foundthe following pattern.Theorem 1.5If g(y) = y3 + α 2y +β 2 = 0, then h(z – =α 3z6 +β 3z3 + γ 3 = 0 a:aprTakeHome.4r.doc 1
  2. 2. Math4R April Take Home Exam Name:_______________ Part I: Polynomial Zeros Questions(1) Demonstrate Theorem 1.1 and Lemma 1.1.1Step 1 Use the given theorem to solve for x: x2 + (1 + i)x + (1 – i) = 0 Graph both solutions as vectors on the Complex Plane.Step 2 Use the first lemma to solve for x: x4 – ix2 + i = 0 Graph all four solutions as vectors on the Complex Plane. a:aprTakeHome.4r.doc 2
  3. 3. Math4R April Take Home Exam Name:_______________ Part I: Polynomial Zeros Questions(2) Investigate how to solve a depressed cubic equation.Step 1 Use Theorem 1.2 to solve for x: x3 = 15x + 4Step 2 Based on Theorem 1.3 this root should be Real. Simplify your answer as much as possible to show that it is Real.Step 3 Use this real root, to find the other two roots.Step 4 Graph y = x3 – 15x – 4. What does this graph tell you about your solutions? a:aprTakeHome.4r.doc 3
  4. 4. Math4R April Take Home Exam Name:_______________ Part I: Polynomial Zeros Questions(3) Let f(x) = x3 – 15x2 + 2x – 5, solve for x: f(x) = 0Step 1 I will Confirm Theorem 1.4 By giving you that g(y) = f(y + 5) = y3 – 73y – 245Step 2 I will Confirm Theorem 1.5 By giving you that h(z) = g(z + = 27z6 – 6615z3 + 389017Step 3 Solve h(z) = 0 for z using Lemma 1.1.2Step 4 Use your value of z to find y = z + , the solution for g(y) = 0.Step 5 Finally, use your value of y to find x = y + 5, the solution for f(x) = 0. a:aprTakeHome.4r.doc 4
  5. 5. Math4R April Take Home Exam Name:_______________ Part II: Polygonal Areas BackgroundTheorem 2Given the vertices of an n-sided polygon, (x0, y0), (x1, y1), (x2, y2), …, xn-1, yn-1), thearea A is given by: A= Note: when i+1 = n, replace i+1 with 0.Lemma 2.1The area of a triangle with vertices (x0, y0), (x1, y1), (x2, y2) is given by: A= A = [(x0 y1 – y0 x1) + (x1 y2 – y1 x2) + (x2 y0 – y2 x0)]This theorem is also known as the Surveyor’s Formula. Surveyors use thisformula to calculate the area of oddly shaped polygonal plots of land quicklyand accurately. a:aprTakeHome.4r.doc 5
  6. 6. Math4R April Take Home Exam Name:_______________ Part II: Polygonal Areas Questions(1) Confirm Lemma 2.1Step 1 Construct ∆ABC such that A(1, 2), B(4, 3) and C(0, 0)Step 2 Use the distance formula to find the length of each side of ∆ABC. a:aprTakeHome.4r.doc 6
  7. 7. Math4R April Take Home Exam Name:_______________ Part II: Polygonal Areas Questions(1) Confirm Lemma 2.1Step 3 Apply Heron’s Formula to find the area of ∆ABC.Step 4 Now try Lemma 2.1 and see if you get the same area. a:aprTakeHome.4r.doc 7
  8. 8. Math4R April Take Home Exam Name:_______________ Part II: Polygonal Areas Questions(1) Confirm Lemma 2.1Step 5 Find xStep 6 How are the calculations in steps 4 & 5 related?Step 7 Does the order of the vector cross product make a difference? a:aprTakeHome.4r.doc 8
  9. 9. Math4R April Take Home Exam Name:_______________ Part II: Polygonal Areas Questions(2) Show that Theorem 2 is based on triangulation and vector cross products!Step 1 Construct the pentagon ABCDE such that A(5,2), B(6, 4), C(4, 5), D(1, 4) and E(2, 2).Step 2 Apply the Surveyor’s Formula to finding the area of the pentagon.Step 3 Find the following vector cross products. x,x,x,x,xStep 4 Find the sum of all these vector cross products. a:aprTakeHome.4r.doc 9
  10. 10. Math4R April Take Home Exam Name:_______________ Part II: Polygonal Areas Questions(2) Show that Theorem 2 is based on triangulation and vector cross products!Step 5 What does this vector sum have to do with the Surveyor’s Formula.Step 6 Some of the vector cross products contain negative components. Why is this significant?Step 7 Research the Shoelace Algorithm online. Recalculate the area of the pentagon using this algorithm. Is this different from the Surveyor’s Formula? a:aprTakeHome.4r.doc 10

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