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Polynomials and roots

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Polynomials and roots

Jeffrey Bivin

Lake Zurich High School

Jeff.bivin@lz95.org

Last Updated: October 26, 2009

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

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

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

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

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

opposite 15 same 85 opposite 225 same 274 opposite 120

F(x) = x5 – 15x4 + 85x3 – 225x2 + 274x – 120

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

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