Question

Simplify the polynomial (2x^3 + x^2 - 4x) - (9x^3 + - 3x^2)

Answer 1

 (2x3-4x2-3x-9)/(x-3) Final result : 2x2 + 2x + 3 Step by step solution :Step  1  :Equation at the end of step  1  : Step  2  :Equation at the end of step  2  : Step  3  : 2x3 - 4x2 - 3x - 9 Simplify —————————————————— x - 3 Checking for a perfect cube :

 3.1    2x3 - 4x2 - 3x - 9  is not a perfect cube 

Trying to factor by pulling out :

 3.2      Factoring:  2x3 - 4x2 - 3x - 9 

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

Group 1:  -3x - 9 
Group 2:  2x3 - 4x2 

Pull out from each group separately :

Group 1:   (x + 3) • (-3)
Group 2:   (x - 2) • (2x2)

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) = 2x3 - 4x2 - 3x - 9
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  2  and the Trailing Constant is  -9. 

 The factor(s) are: 

of the Leading Coefficient :  1,2 
 of the Trailing Constant :  1 ,3 ,9 

 Let us test ....

  P  Q  P/Q  F(P/Q)   Divisor     -1     1      -1.00      -12.00        -1     2      -0.50      -8.75        -3     1      -3.00      -90.00        -3     2      -1.50      -20.25        -9     1      -9.00     -1764.00        -9     2      -4.50      -258.75        1     1      1.00      -14.00        1     2      0.50      -11.25        3     1      3.00      0.00    x - 3      3     2      1.50      -15.75        9     1      9.00      1098.00        9     2      4.50      78.75   

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 
   2x3 - 4x2 - 3x - 9 
can be divided with  x - 3 

Polynomial Long Division :

 3.4    Polynomial Long Division 
Dividing :  2x3 - 4x2 - 3x - 9 
                              ("Dividend")
By         :    x - 3    ("Divisor")

dividend  2x3 - 4x2 - 3x - 9 - divisor * 2x2   2x3 - 6x2     remainder    2x2 - 3x - 9 - divisor * 2x1     2x2 - 6x   remainder      3x - 9 - divisor * 3x0       3x - 9 remainder       0

Quotient :  2x2+2x+3  Remainder:  0 

Trying to factor by splitting the middle term

 3.5     Factoring  2x2+2x+3 

The first term is,  2x2  its coefficient is  2 .
The middle term is,  +2x  its coefficient is  2 .
The last term, "the constant", is  +3 

Step-1 : Multiply the coefficient of the first term by the constant   2 • 3 = 6 

Step-2 : Find two factors of  6  whose sum equals the coefficient of the middle term, which is   2 .

     -6   +   -1   =   -7     -3   +   -2   =   -5     -2   +   -3   =   -5     -1   +   -6   =   -7     1   +   6   =   7     2   +   3   =   5     3   +   2   =   5     6   +   1   =   7

Observation : No two such factors can be found !! 
Conclusion : Trinomial can not be factored

Canceling Out :

 3.6    Cancel out  (x-3)  which appears on both sides of the fraction line.

Final result : 2x2 + 2x + 3

Answer 2

The answer is 2x2 + 2x + 3