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If the roots of \(p(x) = x^3 - 15x^2 + 40x + c\) form an arithmetic sequence, find c.

 Jun 20, 2019
 #1
avatar+104688 
+2

Let the roots be  R,  R + D  , R + 2D

 

So....by Vieta.....

 

R + (R +D) + (R + 2D)  = 15

3R + 3D  = 15

3 (R + D)  = 15

R + D  = 5

D = 5 - R

2D = 10-2R

 

And 

R( R + D) + R(R + 2D) + (R+D)(R+2D)  =  40

R[5 ] + R [R + 10 -2R] + [5] [ R + 10 - 2R ] = 40

5R + R[10 -R] + 5[ 10-R]  = 40

5R + 10R - R^2 + 50 - 5R  = 40

-R^2 + 10R + 50  = 40

-R^2 + 10R + 10  = 0

R^2 - 10R - 10  = 0

R^2 - 10R + 25 =  10 + 25

(R - 5)^2  =  35

R  =   5 + √35       or       R  = 5 - √35

So

If R = 5 + √35                or            If  R = 5 - √35

D = -√35                                            D = √35

 

And   R(R+D)(R + 2D)  =  -c

 

So  either

 

(5 + √35) (5) (5  - √35)   = -c         or       (5- √35) (5) (5 + √35)  = -c

(25 - 35)(5) = -c                                       (25 - 35) (5)  = -c

(-10)(5)  = -c                                            (-10)(5)  = - c

-50 = -c                                                       -50  = -c

50  = c                                                         50  = c

 

So

 

c  = 50

 

 

 

cool cool cool

 Jun 20, 2019
 #2
avatar+1015 
+2

Nice one that is a hard question by the looks of all the work your showing good job! 

cheekycheekycheeky

Nickolas  Jun 20, 2019
 #3
avatar+104688 
+1

I just hope that it is correct...LOL!!!!

 

 

cool cool cool

CPhill  Jun 20, 2019

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