# Solve Approximation X5

### Reformatting the input đầu vào :

Changes made lớn your input đầu vào should not affect the solution: (1): "x4" was replaced by "x^4". 1 more similar replacement(s).

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## Step 1 :

### Polynomial Roots Calculator :

1.1 Find roots (zeroes) of : F(x) = x5-x4-1Polynomial Roots Calculator is a mix of methods aimed at finding values ofxfor which F(x)=0 Rational Roots demo is one of the above mentioned tools.

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It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integersThe Rational Root Theorem states that if a polynomial zeroes for a rational numberP/Q then p is a factor of the Trailing Constant and Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is 1 and the Trailing Constant is -1. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 Let us test ....

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PQP/QF(P/Q)Divisor
-11 -1.00 -3.00
11 1.00 -1.00

Polynomial Roots Calculator found no rational roots

Equation at the kết thúc of step 1 :

x5 - x4 - 1 = 0

## Step 2 :

Equations of order 5 or higher:2.1Solvex5-x4-1 = 0Points regarding equations of degree five or higher.(1) There is no general method (Formula) for solving polynomial equations of degree five or higher.(2) By the Fundamental theorem of Algebra, if we allow complex numbers, an equation of degree n will have exactly n solutions (This is if we count double solutions as 2, triple solutions as 3 and so on)(3) By the Abel-Ruffini theorem, the solutions can not always be presented in the conventional way using only a finite amount of additions, subtractions, multiplications, divisions or root extractions

(4) If F(x) is a polynomial of odd degree with real coefficients, then the equation F(X)=0 has at least one real solution.(5) Using methods such as the Bisection Method, real solutions can be approximated khổng lồ any desired degree of accuracy.

Approximating a root using the Bisection Method :

We now use the Bisection Method lớn approximate one of the solutions. The Bisection Method is an iterative procedure lớn approximate a root (Root is another name for a solution of an equation).The function is F(x) = x5 - x4 - 1Atx= 1.00 F(x) is equal khổng lồ -1.00Atx= 2.00 F(x) is equal to lớn 15.00Intuitively we feel, & justly so, that since F(x) is negative on one side of the interval, và positive on the other side then, somewhere inside this interval, F(x) is zero Procedure :(1) Find a point "Left" where F(Left) (2) Find a point "Right" where F(Right) > 0(3) Compute "Middle" the middle point of the interval (4) Calculate Value = F(Middle)(5) If Value is close enough to lớn zero goto Step (7) Else : If Value then : Left Middle If Value > 0 then : Right Middle(6) Loop back lớn Step (3)(7) Done!! The approximation found is MiddleFollow Middle movements khổng lồ understand how it works :