Lesson 7 - Firstly, what is Vieta’s formula? The idea is...

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Unformatted text preview: Firstly, what is Vieta’s formula? The idea is to be able to expand a any polynomial of the form: ( x- r 1 )( x- r 2 ) · · · ( x- r n ) Where n is the degree of the polynomial or the number of roots. To do this, he used something called Symmetric Sums. S = 1 S 1 = r 1 + r 2 + · · · + r n For j < k S 2 = r 1 r 2 + r 1 r 3 + · · · + r j r k + · · · + r n- 1 r n · · · S n = r 1 r 2 · · · r n This gives : ( x- r 1 )( x- r 2 ) · · · ( x- r n ) = S x n- S 1 x n- 1 + S 2 x n- 2 · · · + (- 1) n S n Enough general theory, here is an example: ( x- 1)( x- 2)( x + 1) S = 1 S 1 = r 1 + r 2 + r 3 = 1 + 2- 1 = 2 S 2 = r 1 r 2 + r 1 r 3 + r 2 r 3 = 1 * 2 + 1 * (- 1) + 2 * (- 1) =- 1 S 3 = r 1 r 2 r 3 = 1 * 2 * (- 1) =- 6 This gives : ( x- 1)( x- 2)( x + 1) = S x 3- S 1 x 2 + S 2 x- S 3 = 1 x 3- 2 x 2 + (- 1) x- (- 6) = x 3- 2 x 2- x + 6 Of course, this idea goes backwards: If we know r 1 + r 2 + r 3 = 3 , r 1 r 2 + r 1 r 3 + r 2 r 3 = 3 , r 1 r 2 r 3 = 1 Then r 1 , r 2 , r 3 are the three roots of the polynomial...
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This note was uploaded on 03/22/2010 for the course MATH 1104 taught by Professor Unknown during the Fall '10 term at Carleton.

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Lesson 7 - Firstly, what is Vieta’s formula? The idea is...

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