## Step 1 :

Checking for a perfect cube :1.1x4-x3-x-1 is not a perfect cube

Trying lớn factor by pulling out :

1.2 Factoring: x4-x3-x-1 Thoughtfully split the expression at hand into groups, each group having two terms:Group 1: -x-1Group 2: x4-x3Pull out from each group separately :Group 1: (x+1) • (-1)Group 2: (x-1) • (x3)Bad news !! Factoring by pulling out fails : The groups have no common factor & can not be added up to size a multiplication.

### Polynomial Roots Calculator :

1.3 Find roots (zeroes) of : F(x) = x4-x3-x-1Polynomial Roots Calculator is a phối of methods aimed at finding values ofxfor 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 integersThe Rational Root Theorem states that if a polynomial zeroes for a rational numberP/Q then p. Is a factor of the Trailing Constant & Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is 1 và the Trailing Constant is -1.

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The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 Let us demo ....

PQP/QF(P/Q)Divisor
-11 -1.00 2.00
11 1.00 -2.00

Polynomial Roots Calculator found no rational roots

Equation at the end of step 1 :

x4 - x3 - x - 1 = 0

## Step 2 :

Quartic Equations:2.1Solvex4-x3-x-1 = 0In search of an interavl at which the above polynomial changes sign, from negative khổng lồ positive or the other wayaround.Method of search: Calculate polynomial values for all integer points between x=-20 & x=+20 Found change of sign between x= -1.00 & x= 0.00

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 to lớn approximate a root (Root is another name for a solution of an equation).The function is F(x) = x4 - x3 - x - 1Atx= 0.00 F(x) is equal to lớn -1.00Atx= -1.00 F(x) is equal to lớn 2.00Intuitively we feel, và 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 to lớn Step (3)(7) Done!! The approximation found is MiddleFollow Middle movements to understand how it works :