Question

W(x) = x2 + 2 x divided by X - 1

Answer 1

Answer:

$x+1+\frac{1}{x-1}$

Step-by-step explanation:

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "x2"   was replaced by   "x^2".  1 more similar replacement(s).

STEP

1

:

Equation at the end of step 1

 

STEP

2

:

           x3 - 2x2 + 2x - 1

Simplify   —————————————————

                 x - 1      

Checking for a perfect cube :

2.1    x3 - 2x2 + 2x - 1  is not a perfect cube

Trying to factor by pulling out :

2.2      Factoring:  x3 - 2x2 + 2x - 1  

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

Group 1:  2x - 1  

Group 2:  -2x2 + x3  

Pull out from each group separately :

Group 1:   (2x - 1) • (1)

Group 2:   (x - 2) • (x2)

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 :

2.3    Find roots (zeroes) of :       F(x) = x3 - 2x2 + 2x - 1

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  -1.

The factor(s) are:

of the Leading Coefficient :  1

of the Trailing Constant :  1

Let us test ....

  P    Q    P/Q    F(P/Q)     Divisor

     -1       1        -1.00        -6.00      

     1       1        1.00        0.00      x - 1  

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

  x3 - 2x2 + 2x - 1  

can be divided with  x - 1  

Polynomial Long Division :

2.4    Polynomial Long Division

Dividing :  x3 - 2x2 + 2x - 1  

                             ("Dividend")

By         :    x - 1    ("Divisor")

dividend     x3  -  2x2  +  2x  -  1  

- divisor  * x2     x3  -  x2          

remainder      -  x2  +  2x  -  1  

- divisor  * -x1      -  x2  +  x      

remainder             x  -  1  

- divisor  * x0             x  -  1  

remainder                0

Quotient :  x2-x+1  Remainder:  0  

Trying to factor by splitting the middle term

2.5     Factoring  x2-x+1  

The first term is,  x2  its coefficient is  1 .

The middle term is,  -x  its coefficient is  -1 .

The last term, "the constant", is  +1  

Step-1 : Multiply the coefficient of the first term by the constant   1 • 1 = 1  

Step-2 : Find two factors of  1  whose sum equals the coefficient of the middle term, which is   -1 .

     -1    +    -1    =    -2  

     1    +    1    =    2  

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Canceling Out :

2.6    Cancel out  (x-1)  which appears on both sides of the fraction line.

Final result :

 x2 - x + 1