x = 1
x =(7-√77)/2=-0.887
x =(7+√77)/2= 7.887
 (((x3) -  23x2) +  17) -  10  = 0
Step  2  :
Polynomial Roots Calculator :
2.1 Â Â Find roots (zeroes) of : Â Â Â F(x) = x3-8x2+7
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which  F(x)=0 Â
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  7.
The factor(s) are:
of the Leading Coefficient : Â 1
of the Trailing Constant : Â 1 ,7
Let us test
 P   Q   P/Q   F(P/Q)   Divisor
   -1    1     -1.00     -2.00  Â
   -7    1     -7.00     -728.00  Â
   1    1     1.00     0.00    x-1
   7    1     7.00     -42.00  Â
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
 x3-8x2+7
can be divided with  x-1
Polynomial Long Division :
2.2 Â Â Polynomial Long Division
Dividing : Â x3-8x2+7
               ("Dividend")
By     :   x-1   ("Divisor")
dividend   x3  -  8x2    +  7
- divisor  * x2   x3  -  x2    Â
remainder    -  7x2    +  7
- divisor  * -7x1    -  7x2  +  7x  Â
remainder      -  7x  +  7
- divisor  * -7x0      -  7x  +  7
remainder         0
Quotient : Â x2-7x-7 Â Remainder: Â 0
Trying to factor by splitting the middle term
2.3   Factoring  x2-7x-7
The first term is,  x2  its coefficient is  1 .
The middle term is,  -7x  its coefficient is  -7 .
The last term, "the constant", is  -7
Step-1 : Multiply the coefficient of the first term by the constant  1 • -7 = -7
Step-2 : Find two factors of  -7  whose sum equals the coefficient of the middle term, which is  -7 .
   -7   +   1   =   -6
   -1   +   7   =   6
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step  2  :
 (x2 - 7x - 7) • (x - 1)  = 0
Step  3  :
Theory - Roots of a product :
3.1 Â Â 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.
Parabola, Finding the Vertex :
3.2    Find the Vertex of  y = x2-7x-7
For any parabola,Ax2+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is  3.5000 Â
Plugging into the parabola formula  3.5000  for  x  we can calculate the  y -coordinate :
 y = 1.0 * 3.50 * 3.50 - 7.0 * 3.50 - 7.0
or  y = -19.250
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : Â y = x2-7x-7
Axis of Symmetry (dashed) Â {x}={ 3.50}
Vertex at  {x,y} = { 3.50,-19.25} Â
x -Intercepts (Roots) :
Root 1 at  {x,y} = {-0.89, 0.00}
Root 2 at  {x,y} = { 7.89, 0.00}
Solve Quadratic Equation by Completing The Square
3.3   Solving  x2-7x-7 = 0 by Completing The Square .
Add  7  to both side of the equation :
 x2-7x = 7
Now the clever bit: Take the coefficient of  x , which is  7 , divide by two, giving  7/2 , and finally square it giving  49/4
Add  49/4  to both sides of the equation :
 On the right hand side we have :
 7  +  49/4   or,  (7/1)+(49/4)
 The common denominator of the two fractions is  4  Adding  (28/4)+(49/4)  gives  77/4
 So adding to both sides we finally get :
 x2-7x+(49/4) = 77/4
Adding  49/4  has completed the left hand side into a perfect square :
 x2-7x+(49/4)  =
 (x-(7/2)) • (x-(7/2))  =
 (x-(7/2))2
Things which are equal to the same thing are also equal to one another. Since
 x2-7x+(49/4) = 77/4 and
 x2-7x+(49/4) = (x-(7/2))2
then, according to the law of transitivity,
 (x-(7/2))2 = 77/4
We'll refer to this Equation as  Eq. #3.3.1 Â
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
 (x-(7/2))2  is
 (x-(7/2))2/2 =
 (x-(7/2))1 =
 x-(7/2)
Now, applying the Square Root Principle to  Eq. #3.3.1  we get:
 x-(7/2) = √ 77/4
Add  7/2  to both sides to obtain:
 x = 7/2 + √ 77/4
Since a square root has two values, one positive and the other negative
 x2 - 7x - 7 = 0
 has two solutions:
 x = 7/2 + √ 77/4
 or
 x = 7/2 - √ 77/4
Note that  √ 77/4 can be written as
 √ 77  / √ 4  which is √ 77  / 2
Solve Quadratic Equation using the Quadratic Formula
3.4   Solving   x2-7x-7 = 0 by the Quadratic Formula .
According to the Quadratic Formula,  x  , the solution for  Ax2+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                 Â
      - B  ±  √ B2-4AC
 x =  ————————
           2A
 In our case,  A  =   1
           B  =   -7
           C  =  -7
Accordingly, Â B2 Â - Â 4AC Â =
          49 - (-28) =
          77
Applying the quadratic formula :
       7 ± √ 77
 x  =   —————
          2
 √ 77  , rounded to 4 decimal digits, is  8.7750
So now we are looking at:
     x  =  ( 7 ±  8.775 ) / 2
Two real solutions:
x =(7+√77)/2= 7.887
or:
x =(7-√77)/2=-0.887
Solving a Single Variable Equation :
3.5    Solve  :   x-1 = 0
Add  1  to both sides of the equation :
           x = 1
x = 1
x =(7-√77)/2=-0.887
x =(7+√77)/2= 7.887
