Suppose that f(x) = x² – 5x + 7 has roots a and B. Use symmetricfunctions to find a + B, aß, a² + B², and a³ + B³.

Answered: = x? – 5x + 7 has roots a and B. Use…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Suppose that f(x) = x² – 5x + 7 has roots a and B. Use symmetric
functions to find a + B, aß, a² + B², and a³ + B³.

Expert Solution

Step 1

If we have a quadratic equation $a x^{2} + b x + c = 0$ and $α$ and $β$ are the roots then:

$α + β = - \frac{b}{a}$

$α β = \frac{c}{a}$

We can use these and get the following:

${(α + β)}^{2} = α^{2} + β^{2} + 2 α β \Rightarrow α^{2} + β^{2} = {(α + β)}^{2} - 2 α β$

And

${(α + β)}^{3} = α^{3} + β^{3} + 3 α β (α + β) \Rightarrow α^{3} + β^{3} = {(α + β)}^{3} - 3 α β (α + β)$

Hence,

$α + β = - \frac{b}{a}$

$α β = \frac{c}{a}$

$α^{2} + β^{2} = {(α + β)}^{2} - 2 α β$

$α^{3} + β^{3} = {(α + β)}^{3} - 3 α β (α + β)$

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= x? – 5x + 7 has roots a and B. Use symmetric Suppose that f(x) functions to find a + B, aß, a² + B², and a³ + B³.