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?
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)
+130085 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
