Step by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "v6" was replaced by "v^6".
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
6*(6*v^6)-5-(1+6*v)=0
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((6 • (2•3v6)) - 5) - (6v + 1) = 0
Step 2 :
Multiplying exponents :
2.1 21 multiplied by 21 = 2(1 + 1) = 22
Multiplying exponents :
2.2 31 multiplied by 31 = 3(1 + 1) = 32
Equation at the end of step 2 :
((22•32v6) - 5) - (6v + 1) = 0
Step 3 :
Step 4 :
Pulling out like terms :
4.1 Pull out like factors :
36v6 - 6v - 6 = 6 • (6v6 - v - 1)
Polynomial Roots Calculator :
4.2 Find roots (zeroes) of : F(v) = 6v6 - v - 1
Polynomial Roots Calculator is a set of methods aimed at finding values of v for which F(v)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers v 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 6 and the Trailing Constant is -1.
The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 6.00 | ||||||
| -1 | 2 | -0.50 | -0.41 | ||||||
| -1 | 3 | -0.33 | -0.66 | ||||||
| -1 | 6 | -0.17 | -0.83 | ||||||
| 1 | 1 | 1.00 | 4.00 | ||||||
| 1 | 2 | 0.50 | -1.41 | ||||||
| 1 | 3 | 0.33 | -1.33 | ||||||
| 1 | 6 | 0.17 | -1.17 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 4 :
6 • (6v6 - v - 1) = 0
Step 5 :
Equations which are never true :
5.1 Solve : 6 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Equations of order 5 or higher :
5.2 Solve 6v6-v-1 = 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 v=-20 and v=+20
Found change of sign between v= -1.00 and v= 0.00
Approximating a root using the Bisection Method :
We now use the Bisection Method to approximate one of the solutions. The Bisection Method is an iterative procedure to approximate a root (Root is another name for a solution of an equation).
The function is F(v) = 6v6 - v - 1
At v= 0.00 F(v) is equal to -1.00
At v= -1.00 F(v) is equal to 6.00
Intuitively we feel, and justly so, that since F(v) is negative on one side of the interval, and positive on the other side then, somewhere inside this interval, F(v) is zero
Procedure :
(1) Find a point "Left" where F(Left) < 0
(2) Find a point 'Right' where F(Right) > 0
(3) Compute 'Middle' the middle point of the interval [Left,Right]
(4) Calculate Value = F(Middle)
(5) If Value is close enough to zero goto Step (7)
Else :
If Value < 0 then : Left <- Middle
If Value > 0 then : Right <- Middle
(6) Loop back to Step (3)
(7) Done!! The approximation found is Middle
Follow Middle movements to understand how it works :
Left Value(Left) Right Value(Right) 0.000000000 -1.000000000 -1.000000000 6.000000000 0.000000000 -1.000000000 -1.000000000 6.000000000 -0.500000000 -0.406250000 -1.000000000 6.000000000 -0.500000000 -0.406250000 -0.750000000 0.817871094 -0.625000000 -0.017372131 -0.750000000 0.817871094 -0.625000000 -0.017372131 -0.687500000 0.321059585 -0.625000000 -0.017372131 -0.656250000 0.135505548 -0.625000000 -0.017372131 -0.640625000 0.055363685 -0.625000000 -0.017372131 -0.632812500 0.018114751 -0.625000000 -0.017372131 -0.628906250 0.000156466 -0.626953125 -0.008660878 -0.628906250 0.000156466 -0.627929688 -0.004265550 -0.628906250 0.000156466 -0.628417969 -0.002057889 -0.628906250 0.000156466 -0.628662109 -0.000951549 -0.628906250 0.000156466 -0.628784180 -0.000397751 -0.628906250 0.000156466 -0.628845215 -0.000120695 -0.628906250 0.000156466 -0.628845215 -0.000120695 -0.628875732 0.000017872 -0.628860474 -0.000051415 -0.628875732 0.000017872
Next Middle will get us close enough to zero:
F( -0.628871918 ) is 0.000000550
The desired approximation of the solution is:
v ≓ -0.628871918
Note, ≓ is the approximation symbol
One solution was found :
v ≓ -0.628871918How did we do?
Please leave us feedback.