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Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. Write the polynomial in standard form.

3, 4+2i, 1+√7

 Feb 7, 2018
 #1
avatar+102408 
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The polynomial will have zeroes of

3, 4 + 2i, 4- 2i, 1 + √7 and 1 - √7

 

So we have

 

(x - 3) (x - (4 + 2i)  )  ( x  - (4 - 2i) )  ( x - (1 - √7)  ) ( x - (1 + √7) )  =

 

Breaking this down, we have

 

( x - (4 + 2i ) (x - ( 4 - 2i)  =  (x^2 -x(4 + 2i) - x (4 - 2i) +  (4 + 2i) (4 - 2i)  =

 

(x^2  -4x - 2ix - 4x + 2ix  +  16 + 4)  =  ( x^2 - 8x + 20)

 

 

And

 

( x - (1 - √7)) ( x - ( 1 + √7)  =  

 

x^2 - x(1 - √7) - x ( 1 + √7) +  (1 - √7) (1 + √7) =

 

x^2  - x + √7x - x - √7x + 1 - 7  =

 

(x^2  - 2x  - 6)

 

 

So  we have

 

(x - 3) ( x^2 - 8x + 20)  (x^2 - 2x - 6)  = 

 

This expands to :

 

x^5 - 13 x^4 + 60 x^3 - 82 x^2 - 144 x + 360

 

 

 

cool cool cool

 Feb 7, 2018

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