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
Explanation: