1. Jeffrey Bivin Lake Zurich High School Jeff.bivin@lz95.org
Polynomials and roots
Jeffrey Bivin
Lake Zurich High School
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: = 50
Find all groups of two of the four numbers and find each product:
1•2 = •3 = •4 = •3 = •4 = •4 = 12
Now add their products: = 35
Find all groups of one of the four numbers and find each product:
Now add their products: = 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 – 175x x – 120
F(x) = x5 – 15x4 + 85x3 – 225x x - 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:
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: = 274
Find all groups of three of the five numbers and find each product:
1•2•3 = •2•4 = •2•5 = •3•4 = •3•5 = 15
1•4•5 = •3•4 = •3•5 = •4•5 = •4•5 = 60
Now add: = 225
Find all groups of two of the five numbers and find each product:
1•2 = •3 = •4 = •5 = •3 = 6
2•4 = •5 = •4 = •5 = •5 = 20
Now add: = 85
Find all groups of one of the five numbers and find each product:
Now add: = 15

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

8. 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) = i •(1 – 2i) = 3 – 6i (1 + 2i)(1 – 2i) = 5
same
Now add: i – 6i = 11
Find all groups of one of the five numbers and find each product:
opposite
Now add: i – 2i = 5
F(x) = x3 – 5x x – 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 )
F(x) = (x – 3)(x2 – 2x + 5)
F(x) = x3 – 2x2 + 5x – 3x2 + 6x – 15
F(x) = x3 – 5x2 + 11x – 15