SOLUTION: Solve 6x⁴ - 35x� + 62x�

SOLUTION: Solve 6x⁴ - 35x� + 62x� - 35x + 6 = 0

Question 1131704: Solve 6x⁴ - 35x� + 62x� - 35x + 6 = 0
Found 4 solutions by rothauserc, MathLover1, ikleyn, MathTherapy:
Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
6x^4 - 35x^3 + 62x^2 - 35x + 6 = 0
:
check for the factors of 6(this is the constant and also the leading coefficient in the above equation)
:
use the Rational Zeros Theorem
:
factors for 6 are 1, 2, 3, 6
:
the possible zeros are
:
(+ or - 1, 2, 3, 6)/(1, 2, 3, 6) = + or - 1, 2, 3, 6, 1/2, 1, 3/2, 3, 1/3, 2/3, 1, 2, 1/6, 1/3, 1/2, 1 =
:
-6, -3, -2, -3/2, -1, -1/2, -2/3, -1/3, -1/6, 1/6, 1/3, 2/3, 1/2, 1, 3/2, 2, 3, 6
:
look at the graph to see if we can eliminate any of the possible zeros
:

:
from the graph we see that there are no negative roots and 4 positive roots
:
we can eliminate 1/6, 2/3, 1, 3/2, 6
:
*******************************************************
the roots are 1/3, 1/2, 2, 3
:
to check this substitute for x in the equation
*******************************************************
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

Solve
....factor completely
write as
and as

...group

solutions:

Answer by ikleyn(52863)   (Show Source): You can put this solution on YOUR website!
.

            This equation of the degree 4 is VERY SPECIAL.
            It relates to the class of so named palindromic equations of the degree 4, which means that its coefficients
            form a palindromic sequence.

            See this Wikipedia article https://en.wikipedia.org/wiki/Reciprocal_polynomial#Palindromic_polynomial

            There is a SPECIAL PROCEDURE in algebra to solve such equations. It is presented below.

= 0 (1) It follows from the equation that x= 0 IS NOT the root. So, we can divide both sides by without loosing the roots. In this way, you will get an equivalent equation = 0. Group and re-write it equivalently in the form - + 50 = 0, or - + 50 = 0. (2) Introduce new variable u = + . Then equation (2) takes a form = 0. Solve this quadratic equation using the quadratic formula = = . The two roots are = = = = and = = = = . Now, to find x, we need to solve two equations a) + = and b) + = . Case a). + = = 0 = = = . So, the two roots are = = 3 and = = . Case b). + = = 0 = = = . So, the two roots are = = 2 and = = . ANSWER. The four roots are , , 2 and 3.

Solved.

The lesson to learn

From this post learn on how to solve palindromic equations. Every palindromic equation of the degree 4 can be solved in this way.

The major steps of the solution are :

a) Divide both sides by ; b) Introduce new variable u = x + ; c) Reduce the equation to a quadratic equation relative new variable u and solve it getting two roots and ; d) Then find x by solving two equations = and = .

Again :

Every palindromic equation of the degree 4 can be solved in this way.

Answer by MathTherapy(10556)   (Show Source): You can put this solution on YOUR website!

Solve 6x⁴ - 35x� + 62x� - 35x + 6 = 0

RATIONAL ROOT THEOREM produces 2 zeroes: 2 and 3. Therefore, 2 of the expression's factors are: (x - 2) and (x - 3).
FOILing these 2 factors results in trinomial: x² - 5x + 6
Now, when 6x⁴ - 35x³ + 62x² - 35x + 6 is divided by x² - 5x + 6, we get: 6x² - 5x + 1. Factoring 6x² - 5x + 1 gives us: (3x - 1)(2x - 1).
Therefore, 6x⁴ - 35x³ + 62x² - 35x + 6 = 0 becomes: (3x - 1)(2x - 1)(x - 2)(x - 3) = 0 and the solutions are:
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SOLUTION: Solve 6x&#8308; - 35x� + 62x� - 35x + 6 = 0

SOLUTION: Solve 6x⁴ - 35x� + 62x� - 35x + 6 = 0