Divide ( 6x³ + 11x² - 19x + 6) by ( 3x - 2 )
Answers
STEP1:
Equation at the end of step 1
((((6•(x3))-11x2)-19x)-6)•(3x-2)
STEP 2 :
Equation at the end of step2:
((((2•3x3) - 11x2) - 19x) - 6) • (3x - 2)
STEP3:
Checking for a perfect cube
6x3-11x2-19x-6 is not a perfect cube
Trying to factor by pulling out :
Factoring: 6x3-11x2-19x-6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -19x-6
Group 2: 6x3-11x2
Pull out from each group separately :
Group 1: (19x+6) • (-1)
Group 2: (6x-11) • (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 :
Find roots (zeroes) of : F(x) = 6x3-11x2-19x-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 6 and the Trailing Constant is -6.
The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,6
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
-1 3 -0.33 -1.11
-1 6 -0.17 -3.17
-2 1 -2.00 -60.00
-2 3 -0.67 0.00 3x+2
-3 1 -3.00 -210.00
-3 2 -1.50 -22.50
-6 1 -6.00 -1584.00
1 1 1.00 -30.00
1 2 0.50 -17.50
1 3 0.33 -13.33
1 6 0.17 -9.44
2 1 2.00 -40.00
2 3 0.67 -21.78
3 1 3.00 0.00 x-3
3 2 1.50 -39.00
6 1 6.00 780.00
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
6x3-11x2-19x-6
can be divided by 3 different polynomials,including by x-3
Polynomial Long Division :
Polynomial long division dividing 6x3-11x2-19x-6 ("Dividend")
By : x-3 ("Divisor")
dividend 6x3 - 11x2 - 19x - 6
- divisor * 6x2 6x3 - 18x2
remainder 7x2 - 19x - 6
- divisor * 7x1 7x2 - 21x
remainder 2x - 6
- divisor * 2x0 2x - 6
remainder 0
Quotient : 6x2+7x+2 Remainder: 0
Trying to factor by splitting the middle term
Factoring 6x2+7x+2
The first term is, 6x2 its coefficient is 6 .
The middle term is, +7x its coefficient is 7 .
The last term, "the constant", is +2
Step-1 : Multiply the coefficient of the first term by the constant 6 • 2 = 12
Step-2 : Find two factors of 12 whose sum equals the coefficient of the middle term, which is 7 .
-12 + -1 = -13
-6 + -2 = -8
-4 + -3 = -7
-3 + -4 = -7
-2 + -6 = -8
-1 + -12 = -13
1 + 12 = 13
2 + 6 = 8
3 + 4 = 7 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 3 and 4
6x2 + 3x + 4x + 2
Step-4 : Add up the first 2 terms, pulling out like factors :
3x • (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 :
(3x+2) • (2x+1)
Which is the desired factorization
Final result :
(2x + 1) • (3x + 2) • (x - 3) • (3x - 2)
oh my god! it took lot of time for me to complete this time
I hope this answer helps you more