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 Dec 8, 2018
 #1
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+2

Solve for x:
x^3 - 5 x^2 - 7 x + 51 = 0

The left hand side factors into a product with two terms:
(x + 3) (x^2 - 8 x + 17) = 0

Split into two equations:
x + 3 = 0 or x^2 - 8 x + 17 = 0

Subtract 3 from both sides:
x = -3 or x^2 - 8 x + 17 = 0

Subtract 17 from both sides:
x = -3 or x^2 - 8 x = -17

Add 16 to both sides:
x = -3 or x^2 - 8 x + 16 = -1

Write the left hand side as a square:
x = -3 or (x - 4)^2 = -1

Take the square root of both sides:
x = -3 or x - 4 = i or x - 4 = -i

Add 4 to both sides:
x = -3 or x = 4 + i or x - 4 = -i

Add 4 to both sides:

x = -3          or x = 4 + i          or x = 4 - i

 Dec 8, 2018
 #2
avatar+100564 
+2

If 4 + i    a root then so is the conjugate 4 - i

 

Unfortunately.....We have to multiply these....!!!!

 

Multiplying these we get the polynomial

 

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

 

x^2 - 8x + 16 - i^2 =

 

x^2 - 8x + 17

 

To find the remaining polynomial....we can perform some polynomial division

 

                         x + 3

x^2 - 8x + 17  [  x^3 - 5x^2 - 7x + 51 ] 

                         x^3 - 8x^2 + 17x

                        ___________________

                                 3x^2 - 24x + 51

                                 3x^2 - 24x + 51

                                ______________

 

Since x + 3  is the remaining polynomial, x = -3  is the remaining root

 

 

cool cool cool

 Dec 8, 2018

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