2x^3 - x^2-13x-6 factorise
Answers
Answer:
(2x + 1) • (x + 2) • (x - 3)
Step-by-step explanation:
STEP
1
:
Equation at the end of step 1
((2x3 - x2) - 13x) - 6
STEP
2
:
Checking for a perfect cube
2.1 2x3-x2-13x-6 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: 2x3-x2-13x-6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -13x-6
Group 2: 2x3-x2
Pull out from each group separately :
Group 1: (13x+6) • (-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-x2-13x-6
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 -6.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 4.00
-1 2 -0.50 0.00 2x+1
-2 1 -2.00 0.00 x+2
-3 1 -3.00 -30.00
-3 2 -1.50 4.50
-6 1 -6.00 -396.00
1 1 1.00 -18.00
1 2 0.50 -12.50
2 1 2.00 -20.00
3 1 3.00 0.00 x-3
3 2 1.50 -21.00
6 1 6.00 312.00
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-x2-13x-6
can be divided by 3 different polynomials,including by x-3
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : 2x3-x2-13x-6
("Dividend")
By : x-3 ("Divisor")
dividend 2x3 - x2 - 13x - 6
- divisor * 2x2 2x3 - 6x2
remainder 5x2 - 13x - 6
- divisor * 5x1 5x2 - 15x
remainder 2x - 6
- divisor * 2x0 2x - 6
remainder 0
Quotient : 2x2+5x+2 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring 2x2+5x+2
The first term is, 2x2 its coefficient is 2 .
The middle term is, +5x its coefficient is 5 .
The last term, "the constant", is +2
Step-1 : Multiply the coefficient of the first term by the constant 2 • 2 = 4
Step-2 : Find two factors of 4 whose sum equals the coefficient of the middle term, which is 5 .
-4 + -1 = -5
-2 + -2 = -4
-1 + -4 = -5
1 + 4 = 5 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 1 and 4
2x2 + 1x + 4x + 2
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (2x+1)
Add up the last 2 terms, pulling out common factors :
2 • (2x+1)
Step-5 : Add up the four terms of step 4 :
(x+2) • (2x+1)
Which is the desired factorization
Final result :
(2x + 1) • (x + 2) • (x - 3)