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Step by Step Solution

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

10*k^6-(6*k-9)=0

Step 1 :

Equation at the end of step 1 :

  (2•5k⁶) -  (6k - 9)  = 0

Step 2 :

Polynomial Roots Calculator :

2.1    Find roots (zeroes) of :       F(k) = 10k⁶-6k+9
Polynomial Roots Calculator is a set of methods aimed at finding values of  k  for which   F(k)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers k which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 10 and the Trailing Constant is 9.

The factor(s) are:

of the Leading Coefficient : 1,2 ,5 ,10
of the Trailing Constant : 1 ,3 ,9

Let us test ....

P	Q	P/Q	F(P/Q)
-1	1	-1.00	25.00
-1	2	-0.50	12.16
-1	5	-0.20	10.20
-1	10	-0.10	9.60
-3	1	-3.00	7317.00

Note - For tidiness, printing of 19 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Equation at the end of step 2 :

  10k⁶ - 6k + 9  = 0

Step 3 :

Equations of order 5 or higher :

3.1 Solve 10k⁶-6k+9 = 0

In search of an interavl at which the above polynomial changes sign, from negative to positive or the other wayaround.

Method of search: Calculate polynomial values for all integer points between k=-20 and k=+20

No interval at which a change of sign occures has been found. Consequently, Bisection Approximation can not be used. As this is a polynomial of an even degree it may not even have any real (as opposed to imaginary) roots