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

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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 – 3x + 2)(x 2 – 7x + 12) F(x) = x 4 – 7x 3 + 12x 2 -3x 3 + 21x 2 - 36x 2x 2 - 14x + 24 F(x) = x 4 – 10x 3 + 35x 2 – 50x + 24

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Jeff Bivin -- LZHS Given the 4 numbers 1, 2, 3, 4 Find the product of the four numbers: 12 3 4 = 24 Find all groups of three of the four numbers and find each product: 12 3 = 612 4 = 813 4 = 1223 4 = 24 Now add their products: 6 + 8 + 12 + 24 = 50 Find all groups of two of the four numbers and find each product: 12 = 2 13 = 3 14 = 4 23 = 6 24 = 8 34 = 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

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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 – 3x + 2)(x 2 – 7x + 12) F(x) = x 4 – 7x 3 + 12x 2 -3x 3 + 21x 2 - 36x 2x 2 - 14x + 24 F(x) = x 4 – 10x 3 + 35x 2 – 50x + 24

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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) -5x 4 + 50x 3 – 175x 2 + 250x – 120 F(x) = (x – 5)(x 4 – 10x 3 + 35x 2 – 50x + 24) F(x) = x 5 – 10x 4 + 35x 3 – 50x 2 + 24x F(x) = x 5 – 15x 4 + 85x 3 – 225x 2 + 274x - 120

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Jeff Bivin -- LZHS Given the 5 numbers 5, 1, 2, 3, 4 Find the product of the five numbers: 5 12 3 4 = 120 Find all groups of four of the five numbers and find each product: 1234 = 241235 = 301245 = 401345 = 60 Now add: 24 + 30 + 40 + 60 + 120 = 274 Find all groups of three of the five numbers and find each product: 123 = 6 124 = 8 125 = 10 134 = 12 135 = 15 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: Now add: 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 +15 + 20 = 85 2345 = 120 145 = 20 234 = 24 235 = 30 245 = 40 345 = 60 12 = 2 13 = 3 14 = 4 15 = 5 23 = 6 24 = 8 25 = 10 34 = 12 35 = 15 45 = 20 Now add: 1 + 2 + 3 + 4 + 5 = 15 Find all groups of one of the five numbers and find each product:

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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 – 15x 4 + 85x 3 – 225x 2 + 274x – 120

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

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Jeff Bivin -- LZHS Write a 3 rd 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 – 4i 2 ) F(x) = (x – 3)(x 2 – 2x + 1 + 4) F(x) = (x – 3)(x 2 – 2x + 5) F(x) = x 3 – 2x 2 + 5x – 3x 2 + 6x – 15 F(x) = x 3 – 5x 2 + 11x – 15

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