What are the quotient and remainder of (3x^4+ 2x^2 - 6x + 1) /(x + 1)

Step-by-step explanation:

(3x4+2x2-6x+1)/(x+1)  

Final result :

 (3x3 + 3x2 + 5x - 1) • (x - 1)

 ——————————————————————————————

             x + 1              

See results of polynomial long division:

1. In step #03.06

Step by step solution :

Step  1  :

Equation at the end of step  1  :

Step  2  :

Equation at the end of step  2  :

Step  3  :

           3x4 + 2x2 - 6x + 1

Simplify   ——————————————————

                 x + 1        

Checking for a perfect cube :

3.1    3x4 + 2x2 - 6x + 1  is not a perfect cube

Trying to factor by pulling out :

3.2      Factoring:  3x4 + 2x2 - 6x + 1  

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

Group 1:  -6x + 1  

Group 2:  3x4 + 2x2  

Pull out from each group separately :

Group 1:   (-6x + 1) • (1) = (6x - 1) • (-1)

Group 2:   (3x2 + 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 :

3.3    Find roots (zeroes) of :       F(x) = 3x4 + 2x2 - 6x + 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  3  and the Trailing Constant is  1.

The factor(s) are:

of the Leading Coefficient :  1,3

of the Trailing Constant :  1

Let us test

 P    Q    P/Q    F(P/Q)    Divisor

     -1       1        -1.00        12.00      

     -1       3        -0.33        3.26      

     1       1        1.00        0.00      x - 1  

     1       3        0.33        -0.74      

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

  3x4 + 2x2 - 6x + 1  

can be divided with  x - 1  

Polynomial Long Division :

3.4    Polynomial Long Division

Dividing :  3x4 + 2x2 - 6x + 1  

                             ("Dividend")

By         :    x - 1    ("Divisor")

dividend     3x4      +  2x2  -  6x  +  1  

- divisor  * 3x3     3x4  -  3x3              

remainder         3x3  +  2x2  -  6x  +  1  

- divisor  * 3x2         3x3  -  3x2          

remainder             5x2  -  6x  +  1  

- divisor  * 5x1             5x2  -  5x      

remainder              -  x  +  1  

- divisor  * -x0              -  x  +  1  

remainder                    0

Quotient :  3x3+3x2+5x-1  Remainder:  0  

Polynomial Roots Calculator :

3.5    Find roots (zeroes) of :       F(x) = 3x3+3x2+5x-1

    See theory in step 3.3

In this case, the Leading Coefficient is  3  and the Trailing Constant is  -1.

The factor(s) are:

of the Leading Coefficient :  1,3

of the Trailing Constant :  1

Let us test

 P    Q    P/Q    F(P/Q)    Divisor

     -1       1        -1.00        -6.00      

     -1       3        -0.33        -2.44      

     1       1        1.00        10.00      

     1       3        0.33        1.11      

Polynomial Roots Calculator found no rational roots

Polynomial Long Division :

3.6    Polynomial Long Division

Dividing :  3x3+3x2+5x-1  

                             ("Dividend")

By         :    x+1    ("Divisor")

dividend     3x3  +  3x2  +  5x  -  1  

- divisor  * 3x2     3x3  +  3x2          

remainder             5x  -  1  

- divisor  * 0x1                  

remainder             5x  -  1  

- divisor  * 5x0             5x  +  5  

remainder              -  6  

Quotient :  3x2+5  

Remainder :  -6  

Final result :

 (3x3 + 3x2 + 5x - 1) • (x - 1)

 ——————————————————————————————

             x + 1