6x^3+5x^2-3x-2by3x-2 divide by long division method
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Answer:
(6x3+5x2-3x-2)/(3x-2)
This deals with finding the roots (zeroes) of polynomials.
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Reformatting the input :
Changes made to your input should not affect the solution:
(1): "x2" was replaced by "x^2". 1 more similar replacement(s).
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Equation at the end of step 1
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Equation at the end of step
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:
6x3 + 5x2 - 3x - 2
Simplify ——————————————————
3x - 2
Checking for a perfect cube :
3.1 6x3 + 5x2 - 3x - 2 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 6x3 + 5x2 - 3x - 2
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -3x - 2
Group 2: 6x3 + 5x2
Pull out from each group separately :
Group 1: (3x + 2) • (-1)
Group 2: (6x + 5) • (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) = 6x3 + 5x2 - 3x - 2
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 -2.
The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1 ,2
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 0.00 x + 1
-1 2 -0.50 0.00 2x + 1
-1 3 -0.33 -0.67
-1 6 -0.17 -1.39
-2 1 -2.00 -24.00
-2 3 -0.67 0.44
1 1 1.00 6.00
1 2 0.50 -1.50
1 3 0.33 -2.22
1 6 0.17 -2.33
2 1 2.00 60.00
2 3 0.67 0.00 3x - 2
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 + 5x2 - 3x - 2
can be divided by 3 different polynomials,including by 3x - 2
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : 6x3 + 5x2 - 3x - 2
("Dividend")
By : 3x - 2 ("Divisor")
dividend 6x3 + 5x2 - 3x - 2
- divisor * 2x2 6x3 - 4x2
remainder 9x2 - 3x - 2
- divisor * 3x1 9x2 - 6x
remainder 3x - 2
- divisor * x0 3x - 2
remainder 0
Quotient : 2x2+3x+1 Remainder: 0
Trying to factor by splitting the middle term
3.5 Factoring 2x2+3x+1
The first term is, 2x2 its coefficient is 2 .
The middle term is, +3x its coefficient is 3 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 2 • 1 = 2
Step-2 : Find two factors of 2 whose sum equals the coefficient of the middle term, which is 3 .
-2 + -1 = -3
-1 + -2 = -3
1 + 2 = 3 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 1 and 2
2x2 + 1x + 2x + 1
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 :
1 • (2x+1)
Step-5 : Add up the four terms of step 4 :
(x+1) • (2x+1)
Which is the desired factorization
Canceling Out :
3.6 Cancel out (3x-2) which appears on both sides of the fraction line.
Final result :
(2x + 1) • (x + 1).
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Hope this helps ☺️☺️
Bye