(k^4_3_3k^3)+(-5k^3+6k^3_8k^5)

Answer:

(k4+3+3k3)+(-5k3+6k3+8k5)

Final result :

 8k5 + k4 + 4k3 + 3

Step by step solution :

Step  1  :

Equation at the end of step  1  :

 (((k4)+3)+(3•(k3)))+(((0-(5•(k3)))+(6•(k3)))+23k5)

Step  2  :

Equation at the end of step  2  :

 (((k4)+3)+(3•(k3)))+(((0-(5•(k3)))+(2•3k3))+23k5)

Step  3  :

Equation at the end of step  3  :

 (((k4)+3)+(3•(k3)))+(((0-5k3)+(2•3k3))+23k5)

Step  4  :

Equation at the end of step  4  :

 (((k4) +  3) +  3k3) +  (8k5 + k3)

Step  5  :

Checking for a perfect cube :

5.1    8k5+k4+4k3+3  is not a perfect cube

Trying to factor by pulling out :

5.2      Factoring:  8k5+k4+4k3+3

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

Group 1:  k4+3

Group 2:  8k5+4k3

Pull out from each group separately :

Group 1:   (k4+3) • (1)

Group 2:   (2k2+1) • (4k3)

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 :

5.3    Find roots (zeroes) of :       F(k) = 8k5+k4+4k3+3

Polynomial Roots Calculator is a set of methods aimed at finding values of  k  for which   F(k)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  k  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  8  and the Trailing Constant is  3.

The factor(s) are:

of the Leading Coefficient :  1,2 ,4 ,8

of the Trailing Constant :  1 ,3

Let us test ....

  P    Q    P/Q    F(P/Q)     Divisor

     -1       1        -1.00        -8.00    

     -1       2        -0.50        2.31    

     -1       4        -0.25        2.93    

     -1       8        -0.13        2.99    

     -3       1        -3.00       -1968.00    

     -3       2        -1.50        -66.19    

     -3       4        -0.75        -0.27    

     -3       8        -0.38        2.75    

     1       1        1.00        16.00    

     1       2        0.50        3.81    

     1       4        0.25        3.07    

     1       8        0.13        3.01    

     3       1        3.00        2136.00    

     3       2        1.50        82.31    

     3       4        0.75        6.90    

     3       8        0.38        3.29    

Polynomial Roots Calculator found no rational roots

Final result :

 8k5 + k4 + 4k3 + 3

Step-by-step explanation: