(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
