Solution - Linear equations with one unknown
Step by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "x7" was replaced by "x^7".
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
x^2-6*x^7-(4*x)=0
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((x2) - (2•3x7)) - 4x = 0
Step 2 :
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
-6x7 + x2 - 4x = -x • (6x6 - x + 4)
Polynomial Roots Calculator :
3.2 Find roots (zeroes) of : F(x) = 6x6 - x + 4
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x 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 4.
The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1 ,2 ,4
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 11.00 | ||||||
| -1 | 2 | -0.50 | 4.59 | ||||||
| -1 | 3 | -0.33 | 4.34 | ||||||
| -1 | 6 | -0.17 | 4.17 | ||||||
| -2 | 1 | -2.00 | 390.00 | ||||||
| -2 | 3 | -0.67 | 5.19 | ||||||
| -4 | 1 | -4.00 | 24584.00 | ||||||
| -4 | 3 | -1.33 | 39.05 | ||||||
| 1 | 1 | 1.00 | 9.00 | ||||||
| 1 | 2 | 0.50 | 3.59 | ||||||
| 1 | 3 | 0.33 | 3.67 | ||||||
| 1 | 6 | 0.17 | 3.83 | ||||||
| 2 | 1 | 2.00 | 386.00 | ||||||
| 2 | 3 | 0.67 | 3.86 | ||||||
| 4 | 1 | 4.00 | 24576.00 | ||||||
| 4 | 3 | 1.33 | 36.38 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 3 :
-x • (6x6 - x + 4) = 0
Step 4 :
Theory - Roots of a product :
4.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Solving a Single Variable Equation :
4.2 Solve : -x = 0
Multiply both sides of the equation by (-1) : x = 0
Equations of order 5 or higher :
4.3 Solve 6x6-x+4 = 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 x=-20 and x=+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
One solution was found :
x = 0How did we do?
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