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Please halp asap! Find a monic quartic polynomial  with rational coefficients whose roots include 2+sqrt5 and 1-sqrt2 . Give your answer in expanded form.

 Jul 22, 2020
 #1
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This is due today. Pleeeease help!

 Jul 23, 2020
 #2
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The polynomial is \(x^4 - 4x^3 + 4x^2 - x + 5\)

 Jul 23, 2020
 #3
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Since  2 + sqrt(5)  is a root  2 - sqrt(5)  is also a root.

This means that  [ x - (2 + sqrt(5)) ]  and  [ x - (2 - sqrt(5)) ]  are factors.

Multiplying these factors together gives you  x2 - 4x - 1.

 

Since  1 - sqrt(2)  is a root  1 + sqrt(2)  is also a root.

This means that  [ x - (1 - sqrt(2)) ]  and  [ x - (1 + sqrt(2)) ]  are factors.

Multiplying these factors together gives you  x2 - 2x - 1.

 

Multiplying  x2 - 4x - 1  and  x2 - 2x - 1  together gives you  x4 - 6x3 + 6x2 + 6x + 1.

 Jul 23, 2020

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