Answer: (x2 - x - 1) • (2x + 1)
Explanation:
((2x3 - x2) - 3x) - 1
STEP
2
:
Checking for a perfect cube
2.1 2x3-x2-3x-1 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: 2x3-x2-3x-1
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -3x-1
Group 2: 2x3-x2
Pull out from each group separately :
Group 1: (3x+1) • (-1)
Group 2: (2x-1) • (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 :
2.3 Find roots (zeroes) of : F(x) = 2x3-x2-3x-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 2 and the Trailing Constant is -1.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -1.00
-1 2 -0.50 0.00 2x+1
1 1 1.00 -3.00
1 2 0.50 -2.50
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-x2-3x-1
can be divided with 2x+1
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : 2x3-x2-3x-1
("Dividend")
By : 2x+1 ("Divisor")
dividend 2x3 - x2 - 3x - 1
- divisor * x2 2x3 + x2
remainder - 2x2 - 3x - 1
- divisor * -x1 - 2x2 - x
remainder - 2x - 1
- divisor * -x0 - 2x - 1
remainder 0
Quotient : x2-x-1 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring x2-x-1
The first term is, x2 its coefficient is 1 .
The middle term is, -x its coefficient is -1 .
The last term, "the constant", is -1
Step-1 : Multiply the coefficient of the first term by the constant 1 • -1 = -1
Step-2 : Find two factors of -1 whose sum equals the coefficient of the middle term, which is -1 .
-1 + 1 = 0
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
(x2 - x - 1) • (2x + 1)