Polynomials and roots Jeffrey Bivin Lake Zurich High

Polynomials and roots Jeffrey Bivin Lake Zurich High School Jeff. bivin@lz 95. org Last October 26, 2009 Jeff Updated: Bivin -- LZHS

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

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 Jeff Bivin -- LZHS

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

Write a 5 th 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)(x 4 – 10 x 3 + 35 x 2 – 50 x + 24) F(x) = x 5 – 10 x 4 + 35 x 3 – 50 x 2 + 24 x -5 x 4 + 50 x 3 – 175 x 2 + 250 x – 120 F(x) = x 5 – 15 x 4 + 85 x 3 – 225 x 2 + 274 x - 120 Jeff Bivin -- LZHS

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 Jeff Bivin -- LZHS

Write a 5 th 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) = x 5 – 15 x 4 + 85 x 3 – 225 x 2 + 274 x – 120 Jeff Bivin -- LZHS

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

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