Solution - Finding the roots of polynomials

Step by step solution :

Step  1  :

            x
 Simplify   —
            3

Equation at the end of step  1  :

    x                         
  ((— • x2) +  (x - 9)3) -  243  = 0 
    3                         

Step  2  :

Multiplying exponential expressions :

 2.1    x¹ multiplied by x² = x^{(1 + 2)} = x³

Equation at the end of step  2  :

   x3                  
  (—— +  (x - 9)3) -  243  = 0 
   3                   

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  3  as the denominator :

                (x - 9)3     (x - 9)3 • 3
    (x - 9)3 =  ————————  =  ————————————
                   1              3      

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 3.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 x3 + (x-9)3 • 3     4x3 - 81x2 + 729x - 2187
 ———————————————  =  ————————————————————————
        3                       3            

Equation at the end of step  3  :

  (4x3 - 81x2 + 729x - 2187)    
  —————————————————————————— -  243  = 0 
              3                 

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  3  as the denominator :

           243     243 • 3
    243 =  ———  =  ———————
            1         3   

Checking for a perfect cube :

 4.2    4x³ - 81x² + 729x - 2187  is not a perfect cube

Trying to factor by pulling out :

 4.3      Factoring:  4x³ - 81x² + 729x - 2187 

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

Group 1:  4x³ - 81x² 
Group 2:  729x - 2187 

Pull out from each group separately :

Group 1:   (4x - 81) • (x²)
Group 2:   (x - 3) • (729)

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 :

 4.4    Find roots (zeroes) of :       F(x) = 4x³ - 81x² + 729x - 2187
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  4  and the Trailing Constant is  -2187.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4
 of the Trailing Constant :  1 ,3 ,9 ,27 ,81 ,243 ,729 ,2187

 Let us test ....

Note - For tidiness, printing of 43 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 4.5       Adding up the two equivalent fractions

 (4x3-81x2+729x-2187) - (243 • 3)     4x3 - 81x2 + 729x - 2916
 ————————————————————————————————  =  ————————————————————————
                3                                3            

Checking for a perfect cube :

 4.6    4x³ - 81x² + 729x - 2916  is not a perfect cube

Trying to factor by pulling out :

 4.7      Factoring:  4x³ - 81x² + 729x - 2916 

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

Group 1:  4x³ - 81x² 
Group 2:  729x - 2916 

Pull out from each group separately :

Group 1:   (4x - 81) • (x²)
Group 2:   (x - 4) • (729)

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 :

 4.8    Find roots (zeroes) of :       F(x) = 4x³ - 81x² + 729x - 2916

     See theory in step 4.4
In this case, the Leading Coefficient is  4  and the Trailing Constant is  -2916.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4
 of the Trailing Constant :  1 ,2 ,3 ,4 ,6 ,9 ,12 ,18 ,27 ,36 , etc

 Let us test ....

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
   4x³ - 81x² + 729x - 2916 
can be divided with  x - 9 

Polynomial Long Division :

 4.9    Polynomial Long Division
Dividing :  4x³ - 81x² + 729x - 2916 
                              ("Dividend")
By         :    x - 9    ("Divisor")

Quotient :  4x²-45x+324  Remainder:  0 

Trying to factor by splitting the middle term

 4.10     Factoring  4x²-45x+324 

The first term is,  4x²  its coefficient is  4 .
The middle term is,  -45x  its coefficient is  -45 .
The last term, "the constant", is  +324 

Step-1 : Multiply the coefficient of the first term by the constant   4 • 324 = 1296 

Step-2 : Find two factors of  1296  whose sum equals the coefficient of the middle term, which is   -45 .

For tidiness, printing of 44 lines which failed to find two such factors, was suppressed

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Equation at the end of step  4  :

  (4x2 - 45x + 324) • (x - 9)
  ———————————————————————————  = 0 
               3             

Step  5  :

When a fraction equals zero :

 5.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

  (4x2-45x+324)•(x-9)
  ——————————————————— • 3 = 0 • 3
           3         

Now, on the left hand side, the  3  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   (4x²-45x+324)  •  (x-9)  = 0

Theory - Roots of a product :

 5.2    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 :

 5.3      Find the Vertex of   y = 4x²-45x+324

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 4 , is positive (greater than zero). 

 Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions. 

 Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex. 

 For any parabola,Ax²+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   5.6250  

 Plugging into the parabola formula   5.6250  for  x  we can calculate the  y -coordinate : 
  y = 4.0 * 5.62 * 5.62 - 45.0 * 5.62 + 324.0
or   y = 197.438

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 4x²-45x+324
Axis of Symmetry (dashed)  {x}={ 5.62} 
Vertex at  {x,y} = { 5.62,197.44} 
Function has no real roots

Solve Quadratic Equation by Completing The Square

 5.4     Solving   4x²-45x+324 = 0 by Completing The Square .

 Divide both sides of the equation by  4  to have 1 as the coefficient of the first term :
   x²-(45/4)x+81 = 0

Subtract  81  from both side of the equation :
   x²-(45/4)x = -81

Now the clever bit: Take the coefficient of  x , which is  45/4 , divide by two, giving  45/8 , and finally square it giving  2025/64 

Add  2025/64  to both sides of the equation :
  On the right hand side we have :
   -81  +  2025/64    or,  (-81/1)+(2025/64) 
  The common denominator of the two fractions is  64   Adding  (-5184/64)+(2025/64)  gives  -3159/64 
  So adding to both sides we finally get :
   x²-(45/4)x+(2025/64) = -3159/64

Adding  2025/64  has completed the left hand side into a perfect square :
   x²-(45/4)x+(2025/64)  =
   (x-(45/8)) • (x-(45/8))  =
  (x-(45/8))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(45/4)x+(2025/64) = -3159/64 and
   x²-(45/4)x+(2025/64) = (x-(45/8))²
then, according to the law of transitivity,
   (x-(45/8))² = -3159/64

We'll refer to this Equation as  Eq. #5.4.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (x-(45/8))²   is
   (x-(45/8))^2/2 =
  (x-(45/8))¹ =
   x-(45/8)

Now, applying the Square Root Principle to  Eq. #5.4.1  we get:
   x-(45/8) = √ -3159/64

Add  45/8  to both sides to obtain:
   x = 45/8 + √ -3159/64
In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 


Since a square root has two values, one positive and the other negative
   x² - (45/4)x + 81 = 0
   has two solutions:
  x = 45/8 + √ 3159/64 •  i 
   or
  x = 45/8 - √ 3159/64 •  i 

Note that  √ 3159/64 can be written as
  √ 3159  / √ 64   which is √ 3159  / 8

Solve Quadratic Equation using the Quadratic Formula

 5.5     Solving    4x²-45x+324 = 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 :
                                     
            - B  ±  √ B²-4AC
  x =   ————————
                      2A

  In our case,  A   =     4
                      B   =   -45
                      C   =  324

Accordingly,  B²  -  4AC   =
                     2025 - 5184 =
                     -3159

Applying the quadratic formula :

               45 ± √ -3159
   x  =    ———————
                        8

In the set of real numbers, negative numbers do not have square roots. A new set of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written  (a+b*i) 

Both   i   and   -i   are the square roots of minus 1

Accordingly,√ -3159  = 
                    √ 3159 • (-1)  =
                    √ 3159  • √ -1   =
                    ±  √ 3159  • i

Can  √ 3159 be simplified ?

Yes!   The prime factorization of  3159   is
   3•3•3•3•3•13 
To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 3159   =  √ 3•3•3•3•3•13   =3•3•√ 39   =
                ±  9 • √ 39

  √ 39   , rounded to 4 decimal digits, is   6.2450
 So now we are looking at:
           x  =  ( 45 ± 9 •  6.245 i ) / 8

Two imaginary solutions :

 x =(45+√-3159)/8=(45+9i√ 39 )/8= 5.6250+7.0256i
  or: 
 x =(45-√-3159)/8=(45-9i√ 39 )/8= 5.6250-7.0256i

Solving a Single Variable Equation :

 5.6      Solve  :    x-9 = 0 

 Add  9  to both sides of the equation : 
                      x = 9

Three solutions were found :

1.  x = 9
2.  x =(45-√-3159)/8=(45-9i√ 39 )/8= 5.6250-7.0256i
3.  x =(45+√-3159)/8=(45+9i√ 39 )/8= 5.6250+7.0256i