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?

Guest Jan 19, 2019

#1**+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)**

Guest Jan 19, 2019

#4**+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

CPhill Jan 19, 2019