Step 1 :

Equation at the end of step 1 : (((2 • (x3)) - 3x2) - 6x) - 9

Step 2 :

Equation at the end of step 2 : ((2x3 - 3x2) - 6x) - 9

Step 3 :

Checking for a perfect cube :3.12x3-3x2-6x-9 is not a perfect cube

Trying to factor by pulling out :

3.2 Factoring: 2x3-3x2-6x-9 Thoughtfully split the expression at hand into groups, each group having two terms:Group 1: -6x-9Group 2: 2x3-3x2Pull out from each group separately :Group 1: (2x+3) • (-3)Group 2: (2x-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 Find roots (zeroes) of : F(x) = 2x3-3x2-6x-9Polynomial Roots Calculator is a set of methods aimed at finding values ofxfor 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 integersThe Rational Root Theorem states that if a polynomial zeroes for a rational numberP/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading CoefficientIn 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 ....

PQP/QF(P/Q)Divisor
-11 -1.00 -8.00
-12 -0.50 -7.00
-31 -3.00 -72.00
-32 -1.50 -13.50
-91 -9.00-1656.00
-92 -4.50 -225.00
11 1.00 -16.00
12 0.50 -12.50
31 3.00 0.00x-3
32 1.50 -18.00
91 9.00 1152.00
92 4.50 85.50

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 2x3-3x2-6x-9can be divided with x-3

Polynomial Long Division :

3.4 Polynomial Long Division Dividing : 2x3-3x2-6x-9("Dividend") By:x-3("Divisor")

dividend2x3-3x2-6x-9
-divisor* 2x22x3-6x2
remainder3x2-6x-9
-divisor* 3x13x2-9x
remainder3x-9
-divisor* 3x03x-9
remainder0

Quotient : 2x2+3x+3 Remainder: 0

Trying to factor by splitting the middle term

3.5Factoring 2x2+3x+3 The first term is, 2x2 its coefficient is 2.The middle term is, +3x its coefficient is 3.The last term, "the constant", is +3Step-1 : Multiply the coefficient of the first term by the consistent 2•3=6Step-2 : Find two factors of 6 whose sum equals the coefficient of the middle term, which is 3.

-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