Find all of the rational zeros of each function.
v(x) = x^3-9x^2 + 27x -27

    x ≓ -0.732511163

Step-by-step explanation: x3-9x2+27x+25=0  

One solution was found :

                        x ≓ -0.732511163

Step by step solution :

Step  1  :

Equation at the end of step  1  :

 (((x3) -  32x2) +  27x) +  25  = 0  

Step  2  :

Checking for a perfect cube :

2.1    x3-9x2+27x+25  is not a perfect cube

Trying to factor by pulling out :

2.2      Factoring:  x3-9x2+27x+25  

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  x3+25  

Group 2:  -9x2+27x  

Pull out from each group separately :

Group 1:   (x3+25) • (1)

Group 2:   (x-3) • (-9x)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

2.3    Find roots (zeroes) of :       F(x) = x3-9x2+27x+25

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  1  and the Trailing Constant is  25.

The factor(s) are:

of the Leading Coefficient :  1

of the Trailing Constant :  1 ,5 ,25

Let us test

 P    Q    P/Q    F(P/Q)    Divisor

     -1       1        -1.00        -12.00      

     -5       1        -5.00        -460.00      

     -25       1       -25.00       -21900.00      

     1       1        1.00        44.00      

     5       1        5.00        60.00      

     25       1        25.00       10700.00      

Polynomial Roots Calculator found no rational roots

Equation at the end of step  2  :

 x3 - 9x2 + 27x + 25  = 0  

Step  3  :

Cubic Equations :

3.1     Solve   x3-9x2+27x+25 = 0

Future releases of Tiger-Algebra will solve equations of the third degree directly.

Meanwhile we will use the Bisection method to approximate one real solution.

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(x) = x3 - 9x2 + 27x + 25

At   x=   -1.00   F(x)  is equal to  -12.00  

At   x=   0.00   F(x)  is equal to  25.00  

Intuitively we feel, and justly so, that since  F(x)  is negative on one side of the interval, and positive on the other side then, somewhere inside this interval,  F(x)  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)

-1.000000000  -12.000000000  0.000000000   25.000000000

-1.000000000  -12.000000000  0.000000000   25.000000000

-1.000000000  -12.000000000 -0.500000000    9.125000000

-0.750000000   -0.734375000 -0.500000000    9.125000000

-0.750000000   -0.734375000 -0.625000000    4.365234375

-0.750000000   -0.734375000 -0.687500000    1.858642578

-0.750000000   -0.734375000 -0.718750000    0.573028564

-0.734375000   -0.077938080 -0.718750000    0.573028564

-0.734375000   -0.077938080 -0.726562500    0.248227596

-0.734375000   -0.077938080 -0.730468750    0.085315526

-0.734375000   -0.077938080 -0.732421875    0.003731437

-0.733398438   -0.037092640 -0.732421875    0.003731437

-0.732910156   -0.016677932 -0.732421875    0.003731437

-0.732666016   -0.006472580 -0.732421875    0.003731437

-0.732543945   -0.001370405 -0.732421875    0.003731437

-0.732543945   -0.001370405 -0.732482910    0.001180558

-0.732513428   -0.000094913 -0.732482910    0.001180558

-0.732513428   -0.000094913 -0.732498169    0.000542825

-0.732513428   -0.000094913 -0.732505798    0.000223957

-0.732513428   -0.000094913 -0.732509613    0.000064522

-0.732511520   -0.000015195 -0.732509613    0.000064522

-0.732511520   -0.000015195 -0.732510567    0.000024663

-0.732511520   -0.000015195 -0.732511044    0.000004734

    Next Middle will get us close enough to zero:

    F( -0.732511163 ) is  -0.000000248  

    The desired approximation of the solution is:

      x ≓ -0.732511163

    Note, ≓ is the approximation symbol

One solution was found :

                        x ≓ -0.732511163

Processing ends successfully hope this helps i tried to break down every thing for you to help you better understand it and you can do it yourself next time

(1,4) is on graph of f(x)

f(1) = 4

so 1/4 f(x)

1/4 f(1)

1/4. ×4

1

so point for function 1/4. f(x) is (1,1)