Solution - Finding the roots of polynomials

Step  1  :

            256
 Simplify   ———
            x2 

Equation at the end of step  1  :

                     256
  ((((x4)-(16•(x2)))-———)+4x)+16
                     x2 
 Step  2  :

Equation at the end of step  2  :

                      256            
  ((((x4) -  24x2) -  ———) +  4x) +  16
                      x2             

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  x²  as the denominator :

                  x4 - 16x2     (x4 - 16x2) • x2
     x4 - 16x2 =  —————————  =  ————————————————
                      1                x2       

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

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   x⁴ - 16x²  =   x² • (x² - 16) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  x² - 16 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 16 is the square of 4
Check :  x²  is the square of  x¹ 

Factorization is :       (x + 4)  •  (x - 4) 

Adding fractions that have a common denominator :

 4.3       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:

 x2 • (x+4) • (x-4) • x2 - (256)     x6 - 16x4 - 256
 ———————————————————————————————  =  ———————————————
               x2                          x2       

Equation at the end of step  4  :

   (x6 - 16x4 - 256)           
  (————————————————— +  4x) +  16
          x2                   

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

          4x     4x • x2
    4x =  ——  =  ———————
          1        x2   

Polynomial Roots Calculator :

 5.2    Find roots (zeroes) of :       F(x) = x⁶ - 16x⁴ - 256
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  -256.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8 ,16 ,32 ,64 ,128 ,256

 Let us test ....

Note - For tidiness, printing of 13 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 5.3       Adding up the two equivalent fractions

 (x6-16x4-256) + 4x • x2     x6 - 16x4 + 4x3 - 256
 ———————————————————————  =  —————————————————————
           x2                         x2          

Equation at the end of step  5  :

  (x6 - 16x4 + 4x3 - 256)    
  ——————————————————————— +  16
            x2               

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

          16     16 • x2
    16 =  ——  =  ———————
          1        x2   

Checking for a perfect cube :

 6.2    x⁶ - 16x⁴ + 4x³ - 256  is not a perfect cube

Trying to factor by pulling out :

 6.3      Factoring:  x⁶ - 16x⁴ + 4x³ - 256 

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

Group 1:  x⁶ - 256 
Group 2:  -16x⁴ + 4x³ 

Pull out from each group separately :

Group 1:   (x⁶ - 256) • (1)
Group 2:   (4x - 1) • (-4x³)

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 :

 6.4    Find roots (zeroes) of :       F(x) = x⁶ - 16x⁴ + 4x³ - 256

     See theory in step 5.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -256.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8 ,16 ,32 ,64 ,128 ,256

 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
   x⁶ - 16x⁴ + 4x³ - 256 
can be divided with  x - 4 

Polynomial Long Division :

 6.5    Polynomial Long Division
Dividing :  x⁶ - 16x⁴ + 4x³ - 256 
                              ("Dividend")
By         :    x - 4    ("Divisor")

Quotient :  x⁵+4x⁴+4x²+16x+64  Remainder:  0 

Polynomial Roots Calculator :

 6.6    Find roots (zeroes) of :       F(x) = x⁵+4x⁴+4x²+16x+64

     See theory in step 5.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  64.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8 ,16 ,32 ,64

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 6.7       Adding up the two equivalent fractions

 (x5+4x4+4x2+16x+64) • (x-4) + 16 • x2      x6 - 16x4 + 4x3 + 16x2 - 256 
 —————————————————————————————————————  =  ————————————————————————————
                  x2                                    x2             

Polynomial Roots Calculator :

 6.8    Find roots (zeroes) of :       F(x) = x⁶ - 16x⁴ + 4x³ + 16x² - 256

     See theory in step 5.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -256.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8 ,16 ,32 ,64 ,128 ,256

 Let us test ....

Note - For tidiness, printing of 13 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Final result :

  x6  16x4 + 4x3 + 16x2 - 256 
  ————————————————————————————
               x2