81 = X + Y and $2 = XY be the elementary symmetric polynomials in R[X, Y], and for each d > 1, let Pa(X,Y)= Xª+yd (a) Prove that p₁ = 81, p2 = s1 - 282, and p3 = s³ -38182. (b) Prove by induction that Pa E R[81, 82] for all d > 1. (Hint. What does pa - si look like?)
81 = X + Y and $2 = XY be the elementary symmetric polynomials in R[X, Y], and for each d > 1, let Pa(X,Y)= Xª+yd (a) Prove that p₁ = 81, p2 = s1 - 282, and p3 = s³ -38182. (b) Prove by induction that Pa E R[81, 82] for all d > 1. (Hint. What does pa - si look like?)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 56E
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![7.27. Let
$1 = X + Y and
82 = XY
be the elementary symmetric polynomials in R[X, Y], and for each d > 1, let
Pa (X,Y)= Xd + yd.
- 282, and p3= s³ -38182.
(a) Prove that p₁ = 81, P2 =s
(b) Prove by induction that
Pa E R[81, 82] for all d > 1.
(Hint. What does pa si look like?)
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc98c3d06-b4a0-47ab-9898-9b16a61ca414%2F96f50e1e-b7db-4867-b074-607dd64c5335%2Flkzk02k_processed.png&w=3840&q=75)
Transcribed Image Text:7.27. Let
$1 = X + Y and
82 = XY
be the elementary symmetric polynomials in R[X, Y], and for each d > 1, let
Pa (X,Y)= Xd + yd.
- 282, and p3= s³ -38182.
(a) Prove that p₁ = 81, P2 =s
(b) Prove by induction that
Pa E R[81, 82] for all d > 1.
(Hint. What does pa si look like?)
-
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