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Other Factorizations

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We think you wrote:

40-(3+2)x6

This solution deals with other factorizations.

Solution found

-5*(x^6-8)
-5*(x^6-8)

Step by Step Solution

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Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x6"   was replaced by   "x^6". 

Step  1  :

Equation at the end of step  1  :

  40 -  5x6

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   40 - 5x6  =   -5 • (x6 - 8) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  x6 - 8 

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 : 8 is not a square !!

Ruling : Binomial can not be factored as the difference of two perfect squares.

Polynomial Roots Calculator :

 3.3    Find roots (zeroes) of :       F(x) = x6 - 8
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  -8.

 
The factor(s) are:

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

 
Let us test ....

  P  Q  P/Q  F(P/Q)   Divisor
     -1     1      -1.00      -7.00   
     -2     1      -2.00      56.00   
     -4     1      -4.00      4088.00   
     -8     1      -8.00     262136.00   
     1     1      1.00      -7.00   
     2     1      2.00      56.00   
     4     1      4.00      4088.00   
     8     1      8.00     262136.00   


Polynomial Roots Calculator found no rational roots

Final result :

  -5 • (x6 - 8)

Why learn this

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