Question

What is the result when 3x^4+6x^3-2x^2-3x+2 is divided by x+1?

Answer 1

Answer:(3x3 + 3x2 + 5x - 1) • (x - 1)

 ——————————————————————————————

             x + 1            

Step-by-step explanation:Step by Step Solution:

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STEP

1

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Equation at the end of step 1

 

STEP

2

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Equation at the end of step

2

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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