Step-by-step explanation:
STEP
1
:
Equation at the end of step 1
Â
STEP Â
2
:
Equation at the end of step
2
:
Â
STEP
3
:
      10x3 - 3x2 - 7x + 3
Simplify  ———————————————————
         2x - 1    Â
Checking for a perfect cube :
3.1 Â Â 10x3 - 3x2 - 7x + 3 Â is not a perfect cube
Trying to factor by pulling out :
3.2 Â Â Â Factoring: Â 10x3 - 3x2 - 7x + 3 Â
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: Â -7x + 3 Â
Group 2: Â 10x3 - 3x2 Â
Pull out from each group separately :
Group 1:  (-7x + 3) • (1) = (7x - 3) • (-1)
Group 2:  (10x - 3) • (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 Â Â Find roots (zeroes) of : Â Â Â F(x) = 10x3 - 3x2 - 7x + 3
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which  F(x)=0 Â
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  10  and the Trailing Constant is  3.
The factor(s) are:
of the Leading Coefficient : Â 1,2 ,5 ,10
of the Trailing Constant : Â 1 ,3
Let us test
 P   Q   P/Q   F(P/Q)   Divisor
   -1    1     -1.00     -3.00   Â
   -1    2     -0.50     4.50   Â
   -1    5     -0.20     4.20   Â
   -1    10     -0.10     3.66   Â
   -3    1     -3.00     -273.00   Â
   -3    2     -1.50     -27.00   Â
   -3    5     -0.60     3.96   Â
   -3    10     -0.30     4.56   Â
   1    1     1.00     3.00   Â
   1    2     0.50     0.00    2x - 1 Â
   1    5     0.20     1.56   Â
   1    10     0.10     2.28   Â
   3    1     3.00     225.00   Â
   3    2     1.50     19.50   Â
   3    5     0.60     -0.12   Â
   3    10     0.30     0.90   Â
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
 10x3 - 3x2 - 7x + 3 Â
can be divided with  2x - 1 Â
Polynomial Long Division :
3.4 Â Â Polynomial Long Division
Dividing : Â 10x3 - 3x2 - 7x + 3 Â
               ("Dividend")
By     :   2x - 1   ("Divisor")
dividend   10x3  -  3x2  -  7x  +  3 Â
- divisor  * 5x2   10x3  -  5x2     Â
remainder     2x2  -  7x  +  3 Â
- divisor  * x1     2x2  -  x   Â
remainder      -  6x  +  3 Â
- divisor  * -3x0      -  6x  +  3 Â
remainder         0
Quotient : Â 5x2+x-3 Â Remainder: Â 0 Â
Trying to factor by splitting the middle term
3.5   Factoring  5x2+x-3 Â
The first term is,  5x2  its coefficient is  5 .
The middle term is,  +x  its coefficient is  1 .
The last term, "the constant", is  -3 Â
Step-1 : Multiply the coefficient of the first term by the constant  5 • -3 = -15 Â
Step-2 : Find two factors of  -15  whose sum equals the coefficient of the middle term, which is  1 .
   -15   +   1   =   -14 Â
   -5   +   3   =   -2 Â
   -3   +   5   =   2 Â
   -1   +   15   =   14 Â
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Canceling Out :
3.6   Cancel out  (2x-1)  which appears on both sides of the fraction line.
Final result :
 5x2 + x - 3