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The polynomial P(x) = 3x^3 - 17x^2 + 26x - 10 has a root at x = 5/3, What is the largest root of P(x)?

 

I was confused after I simplified x = 5/3 to 3x - 5 = 0. Then I divided P(x) by 3x-5 and  got x^2+4x+15 remainder 65. I knew the quadratic formula wasn't going to help so I was confused to what to do. How would you solve this equation?

 Jan 19, 2019
 #1
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+1

Solve for x:
3 x^3 - 17 x^2 + 26 x - 10 = 0

The left hand side factors into a product with two terms:
(3 x - 5) (x^2 - 4 x + 2) = 0

Split into two equations:
3 x - 5 = 0 or x^2 - 4 x + 2 = 0

Add 5 to both sides:
3 x = 5 or x^2 - 4 x + 2 = 0

Divide both sides by 3:
x = 5/3 or x^2 - 4 x + 2 = 0

Subtract 2 from both sides:
x = 5/3 or x^2 - 4 x = -2

Add 4 to both sides:
x = 5/3 or x^2 - 4 x + 4 = 2

Write the left hand side as a square:
x = 5/3 or (x - 2)^2 = 2

Take the square root of both sides:
x = 5/3 or x - 2 = sqrt(2) or x - 2 = -sqrt(2)

Add 2 to both sides:
x = 5/3 or x = 2 + sqrt(2) or x - 2 = -sqrt(2)

Add 2 to both sides:

x = 5/3     or      x = 2 + sqrt(2)      or      x = 2 - sqrt(2)

 Jan 19, 2019
 #2
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Wait but if you use quadratic formula it simplifies out to 2+-2*sqrt(2) all over 2. so the two 2's would cancel out and you would be left with:

x = 5/3 or x = sqrt 2 or x = -sqrt 2

Guest Jan 19, 2019
 #3
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This  is a "cubic equation" not a "qudratic equation".
You could use the "cubic formula" which is analogous to "quadratic formula", which will give you the same results:
x = 3.41421 3562     =2 + sqrt(2)
x = 0.58578 64376   =2 -  sqrt(2)
x = 1.66666 6667     =5/3

Guest Jan 19, 2019
 #4
avatar+107405 
+1

We can perform some synthetic division

 

5/3    [   3    -17       26      - 10      ]

                      5     - 20       10 

           _____________________

            3     -12        6          0

 

The remaining polynomial  is

 

3x^2 - 12x + 6           set this to 0

 

3x^2 - 12x + 6 =  0

 

x^2 - 4x + 2   =   0         complete the square on x

 

x^2 - 4x + 4    =  -2 + 4

 

(x - 2)^2  = 2          take the positive root

 

x - 2 = √2

 

x = 2 + √2   .......the largest root

 

 

 

 

cool cool cool

 Jan 19, 2019

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