Step-by-step explanation:
STEP
1
:
      10
Simplify  ——
      x
Equation at the end of step
1
:
            10
 x4)-(5•(x2)))-6x)-——)-3
            x
STEP
2
:
Equation at the end of step
2
:
              10  Â
 x4) -  5x2) -  6x) -  ——) -  3
              x   Â
STEP
3
:
Rewriting the whole as an Equivalent Fraction
3.1 Â Subtracting a fraction from a whole
Rewrite the whole as a fraction using  x  as the denominator :
           x4 - 5x2 - 6x   (x4 - 5x2 - 6x) • x
  x4 - 5x2 - 6x =  —————————————  =  ———————————————————
              1           x    Â
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 Â Â Pull out like factors :
 x4 - 5x2 - 6x  =  x • (x3 - 5x - 6)
Polynomial Roots Calculator :
4.2 Â Â Find roots (zeroes) of : Â Â Â F(x) = x3 - 5x - 6
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  -6.
The factor(s) are:
of the Leading Coefficient : Â 1
of the Trailing Constant : Â 1 ,2 ,3 ,6
Let us test
 P   Q   P/Q   F(P/Q)   Divisor
   -1    1     -1.00     -2.00  Â
   -2    1     -2.00     -4.00  Â
   -3    1     -3.00     -18.00  Â
   -6    1     -6.00     -192.00  Â
   1    1     1.00     -10.00  Â
   2    1     2.00     -8.00  Â
   3    1     3.00     6.00  Â
   6    1     6.00     180.00  Â
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
4.3 Â Â Â 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:
x • (x3-5x-6) • x - (10)   x5 - 5x3 - 6x2 - 10
————————————————————————  =  ———————————————————
      x              x    Â
Equation at the end of step
4
:
 (x5 - 5x3 - 6x2 - 10)  Â
 ————————————————————— -  3
      x       Â
STEP
5
:
Rewriting the whole as an Equivalent Fraction
5.1 Â Subtracting a whole from a fraction
Rewrite the whole as a fraction using  x  as the denominator :
    3   3 • x
  3 =  —  =  —————
    1    x Â
Checking for a perfect cube :
5.2 Â Â x5 - 5x3 - 6x2 - 10 Â is not a perfect cube
Trying to factor by pulling out :
5.3 Â Â Â Factoring: Â x5 - 5x3 - 6x2 - 10
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: Â -6x2 - 10
Group 2: Â x5 - 5x3
Pull out from each group separately :
Group 1:  (3x2 + 5) • (-2)
Group 2:  (x2 - 5) • (x3)
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 :
5.4 Â Â Find roots (zeroes) of : Â Â Â F(x) = x5 - 5x3 - 6x2 - 10
  See theory in step 4.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -10.
The factor(s) are:
of the Leading Coefficient : Â 1
of the Trailing Constant : Â 1 ,2 ,5 ,10
Let us test
 P   Q   P/Q   F(P/Q)   Divisor
   -1    1     -1.00     -12.00  Â
   -2    1     -2.00     -26.00  Â
   -5    1     -5.00    -2660.00  Â
   -10    1    -10.00    -95610.00  Â
   1    1     1.00     -20.00  Â
   2    1     2.00     -42.00  Â
   5    1     5.00     2340.00  Â
   10    1     10.00    94390.00  Â
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
5.5 Â Â Â Adding up the two equivalent fractions
(x5-5x3-6x2-10) - (3 • x)    x5 - 5x3 - 6x2 - 3x - 10
—————————————————————————  =  ————————————————————————
      x               x      Â
Polynomial Roots Calculator :
5.6 Â Â Find roots (zeroes) of : Â Â Â F(x) = x5 - 5x3 - 6x2 - 3x - 10
  See theory in step 4.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -10.
The factor(s) are:
of the Leading Coefficient : Â 1
of the Trailing Constant : Â 1 ,2 ,5 ,10
Let us test
 P   Q   P/Q   F(P/Q)   Divisor
   -1    1     -1.00     -9.00  Â
   -2    1     -2.00     -20.00  Â
   -5    1     -5.00    -2645.00  Â
   -10    1    -10.00    -95580.00  Â
   1    1     1.00     -23.00  Â
   2    1     2.00     -48.00  Â
   5    1     5.00     2325.00  Â
   10    1     10.00    94360.00  Â
Polynomial Roots Calculator found no rational roots
Final result :
 x5 - 5x3 - 6x2 - 3x - 10
 ————————————————————————
      x      Â