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Polynomials and roots

This document outlines the process of constructing polynomials based on specified roots, including both 4th and 5th degree polynomials. It illustrates how to factor polynomials using roots, calculate product sums for groups of numbers, and derive polynomial expressions by combining factors. The examples provided demonstrate working with real roots as well as complex roots, exemplifying polynomial construction and evaluating group products for various combinations. This resource is beneficial for high school students learning polynomial functions and their properties. ###

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Polynomials and roots

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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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: 2•3•4•5 = 120 1•2•3•4 = 24 1•2•3•5 = 30 1•2•4•5 = 40 1•3•4•5 = 60 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

  7. 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

  8. Given the 5 numbers 3, 1±2i 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

  9. 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

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