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Math4R                April Take Home Exam                    Name:_______________


                         Part I: Polynomial Zeros Background

Let’s solve Quadratic Equations with Complex coefficients using the
Quadratic Formula!

Theorem 1.1
Let f(x) = x2 + α x + β where α , β ∈ C. If f(x) = 0, then x = ∈ C.

Lemma 1.1.1
If f(x2) = x4 + α x2 + β where α , β ∈ C, then x = ∈ C.

Lemma 1.1.2
If f(x3) = x6 + α x3 + β where α , β ∈ C, then x = ∈ C.

Scipione del Ferro (1465 – 1526) discovered the following Cubic Formula.

Theorem 1.2
One 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 1.3
The 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 ∈ ℜ

Then Girolamo Cardano (1501 – 1576) stated the following.

Theorem 1.4
If 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 Cubic
Equations. Further, Cardan’s Theorem allows us to “depress” any Cubic
Equation. Now we can solve any Cubic Equation! How does all this work?
Well, its really the Quadratic Formula in disguise since Cardano also found
the following pattern.

Theorem 1.5
If g(y) = y3 + α 2y +β 2 = 0, then h(z – =α 3z6 +β 3z3 + γ 3 = 0




                                a:aprTakeHome.4r.doc                                      1
Math4R              April Take Home Exam               Name:_______________


                        Part I: Polynomial Zeros Questions

(1) Demonstrate Theorem 1.1 and Lemma 1.1.1
Step 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
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 + 4
Step 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
Math4R                April Take Home Exam                Name:_______________


                          Part I: Polynomial Zeros Questions

(3) Let f(x) = x3 – 15x2 + 2x – 5, solve for x: f(x) = 0
Step 1         I will Confirm Theorem 1.4
               By giving you that g(y) = f(y + 5) = y3 – 73y – 245
Step 2         I will Confirm Theorem 1.5
               By giving you that h(z) = g(z + = 27z6 – 6615z3 + 389017
Step 3         Solve h(z) = 0 for z using Lemma 1.1.2
Step 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
Math4R               April Take Home Exam               Name:_______________


                       Part II: Polygonal Areas Background

Theorem 2
Given the vertices of an n-sided polygon, (x0, y0), (x1, y1), (x2, y2), …, xn-1, yn-1), the
area A is given by:
                                      A=
                     Note: when i+1 = n, replace i+1 with 0.

Lemma 2.1
The 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 this
formula to calculate the area of oddly shaped polygonal plots of land quickly
and accurately.




                              a:aprTakeHome.4r.doc                                       5
Math4R              April Take Home Exam             Name:_______________


                       Part II: Polygonal Areas Questions

(1) Confirm Lemma 2.1
Step 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
Math4R             April Take Home Exam            Name:_______________


                       Part II: Polygonal Areas Questions

(1) Confirm Lemma 2.1
Step 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
Math4R              April Take Home Exam              Name:_______________


                       Part II: Polygonal Areas Questions

(1) Confirm Lemma 2.1
Step 5       Find x
Step 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
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,x
Step 4       Find the sum of all these vector cross products.




                            a:aprTakeHome.4r.doc                                    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 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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LApreC2010-106matrixaprtakehome_4r

  • 1. Math4R April Take Home Exam Name:_______________ Part I: Polynomial Zeros Background Let’s solve Quadratic Equations with Complex coefficients using the Quadratic Formula! Theorem 1.1 Let f(x) = x2 + α x + β where α , β ∈ C. If f(x) = 0, then x = ∈ C. Lemma 1.1.1 If f(x2) = x4 + α x2 + β where α , β ∈ C, then x = ∈ C. Lemma 1.1.2 If f(x3) = x6 + α x3 + β where α , β ∈ C, then x = ∈ C. Scipione del Ferro (1465 – 1526) discovered the following Cubic Formula. Theorem 1.2 One 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 1.3 The 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 ∈ ℜ Then Girolamo Cardano (1501 – 1576) stated the following. Theorem 1.4 If 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 Cubic Equations. Further, Cardan’s Theorem allows us to “depress” any Cubic Equation. Now we can solve any Cubic Equation! How does all this work? Well, its really the Quadratic Formula in disguise since Cardano also found the following pattern. Theorem 1.5 If g(y) = y3 + α 2y +β 2 = 0, then h(z – =α 3z6 +β 3z3 + γ 3 = 0 a:aprTakeHome.4r.doc 1
  • 2. Math4R April Take Home Exam Name:_______________ Part I: Polynomial Zeros Questions (1) Demonstrate Theorem 1.1 and Lemma 1.1.1 Step 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. 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 + 4 Step 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. Math4R April Take Home Exam Name:_______________ Part I: Polynomial Zeros Questions (3) Let f(x) = x3 – 15x2 + 2x – 5, solve for x: f(x) = 0 Step 1 I will Confirm Theorem 1.4 By giving you that g(y) = f(y + 5) = y3 – 73y – 245 Step 2 I will Confirm Theorem 1.5 By giving you that h(z) = g(z + = 27z6 – 6615z3 + 389017 Step 3 Solve h(z) = 0 for z using Lemma 1.1.2 Step 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. Math4R April Take Home Exam Name:_______________ Part II: Polygonal Areas Background Theorem 2 Given the vertices of an n-sided polygon, (x0, y0), (x1, y1), (x2, y2), …, xn-1, yn-1), the area A is given by: A= Note: when i+1 = n, replace i+1 with 0. Lemma 2.1 The 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 this formula to calculate the area of oddly shaped polygonal plots of land quickly and accurately. a:aprTakeHome.4r.doc 5
  • 6. Math4R April Take Home Exam Name:_______________ Part II: Polygonal Areas Questions (1) Confirm Lemma 2.1 Step 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. Math4R April Take Home Exam Name:_______________ Part II: Polygonal Areas Questions (1) Confirm Lemma 2.1 Step 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. Math4R April Take Home Exam Name:_______________ Part II: Polygonal Areas Questions (1) Confirm Lemma 2.1 Step 5 Find x Step 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. 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,x Step 4 Find the sum of all these vector cross products. a:aprTakeHome.4r.doc 9
  • 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