Jeffrey Bivin Lake Zurich High School Jeff.bivin@lz95.org Polynomials and roots Jeffrey Bivin Lake Zurich High School Jeff.bivin@lz95.org Last Updated: October 26, 2009

Write a 4th degree polynomial with the given the roots of 1, 2, 3, 4 F(x) = (x – 1)(x – 2)(x – 3)(x – 4) F(x) = (x2 – 3x + 2)(x2 – 7x + 12) F(x) = x4 – 7x3 + 12x2 -3x3 + 21x2 - 36x 2x2 - 14x + 24 F(x) = x4 – 10x3 + 35x2 – 50x + 24

Given the 4 numbers 1, 2, 3, 4 Find the product of the four numbers: 1•2•3•4 = 24 Find all groups of three of the four numbers and find each product: 1•2•3 = 6 1•2•4 = 8 1•3•4 = 12 2•3•4 = 24 Now add their products: 6 + 8 + 12 + 24 = 50 Find all groups of two of the four numbers and find each product: 1•2 = 2 1•3 = 3 1•4 = 4 2•3 = 6 2•4 = 8 3•4 = 12 Now add their products: 2 + 3 + 4 + 6 + 8 + 12 = 35 Find all groups of one of the four numbers and find each product: Now add their products: 1+ 2 + 3 + 4 = 10

Write a 4th degree polynomial with the given the roots of 1, 2, 3, 4 F(x) = (x – 1)(x – 2)(x – 3)(x – 4) F(x) = (x2 – 3x + 2)(x2 – 7x + 12) F(x) = x4 – 7x3 + 12x2 -3x3 + 21x2 - 36x 2x2 - 14x + 24 opposite opposite same same F(x) = x4 – 10x3 + 35x2 – 50x + 24

Write a 5th degree polynomial with the given the roots of 5, 1, 2, 3, 4 F(x) = (x - 5)(x – 1)(x – 2)(x – 3)(x – 4) F(x) = (x – 5)(x4 – 10x3 + 35x2 – 50x + 24) F(x) = x5 – 10x4 + 35x3 – 50x2 + 24x -5x4 + 50x3 – 175x2 + 250x – 120 F(x) = x5 – 15x4 + 85x3 – 225x2 + 274x - 120

Given the 5 numbers 5, 1, 2, 3, 4 Find the product of the five numbers: 5•1•2•3•4 = 120 Find all groups of four of the five numbers and find each product: 1•2•3•4 = 24 1•2•3•5 = 30 1•2•4•5 = 40 1•3•4•5 = 60 2•3•4•5 = 120 Now add: 24 + 30 + 40 + 60 + 120 = 274 Find all groups of three of the five numbers and find each product: 1•2•3 = 6 1•2•4 = 8 1•2•5 = 10 1•3•4 = 12 1•3•5 = 15 1•4•5 = 20 2•3•4 = 24 2•3•5 = 30 2•4•5 = 40 3•4•5 = 60 Now add: 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 = 225 Find all groups of two of the five numbers and find each product: 1•2 = 2 1•3 = 3 1•4 = 4 1•5 = 5 2•3 = 6 2•4 = 8 2•5 = 10 3•4 = 12 3•5 = 15 4•5 = 20 Now add: 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 +15 + 20 = 85 Find all groups of one of the five numbers and find each product: Now add: 1 + 2 + 3 + 4 + 5 = 15

Write a 5th degree polynomial with the given the roots of 5, 1, 2, 3, 4 opposite 15 same 85 opposite 225 same 274 opposite 120 F(x) = x5 – 15x4 + 85x3 – 225x2 + 274x – 120

Given the 5 numbers 3, 1±2i opposite same opposite Find the product of the three numbers: opposite 3(1+2i)(1-2i) = 3(1 - 4i2) = 3(1 + 4) = 3(5) = 15 Find all groups of two of the five numbers and find each product: 3•(1 + 2i) = 3 + 6i 3•(1 – 2i) = 3 – 6i (1 + 2i)(1 – 2i) = 5 same Now add: 3 + 6i + 3 – 6i + 5 = 11 Find all groups of one of the five numbers and find each product: opposite Now add: 3 + 1 + 2i + 1 – 2i = 5 F(x) = x3 – 5x2 + 11x – 15

Write a 3rd degree polynomial with the given the roots of 3, 1±2i F(x) = (x – 3)(x – (1+2i))(x – (1–2i)) F(x) = (x – 3)(x – 1 – 2i)(x – 1 + 2i) F(x) = (x – 3)((x – 1) – 2i)((x – 1) + 2i) F(x) = (x – 3)((x – 1)2 – 4i2) F(x) = (x – 3)(x2 – 2x + 1 + 4) F(x) = (x – 3)(x2 – 2x + 5) F(x) = x3 – 2x2 + 5x – 3x2 + 6x – 15 F(x) = x3 – 5x2 + 11x – 15