Question 1131704: Solve 6x⁴ - 35x³ + 62x² - 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)
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use the Rational Zeros Theorem
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factors for 6 are 1, 2, 3, 6
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the possible zeros are
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(+ 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
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look at the graph to see if we can eliminate any of the possible zeros
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from the graph we see that there are no negative roots and 4 positive roots
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we can eliminate 1/6, 2/3, 1, 3/2, 6
:
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the roots are 1/3, 1/2, 2, 3
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to check this substitute for x in the equation
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Answer by MathLover1(20850) (Show Source): Answer by ikleyn(52847) (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(10555) (Show Source):
You can put this solution on YOUR website!
Solve 6x⁴ - 35x³ + 62x² - 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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