simplify (2x^3-3x^2+x-6)/ (x-2)
Answers
Step-by-step explanation:
STEP
1
:
Equation at the end of step 1
((((2•(x3))-3x2)+x)-6)•(x-2)
STEP
2
:
Equation at the end of step
2
:
(((2x3 - 3x2) + x) - 6) • (x - 2)
STEP
3
:
Checking for a perfect cube
3.1 2x3-3x2+x-6 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 2x3-3x2+x-6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x-6
Group 2: 2x3-3x2
Pull out from each group separately :
Group 1: (x-6) • (1)
Group 2: (2x-3) • (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) = 2x3-3x2+x-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 -12.00
-1 2 -0.50 -7.50
-2 1 -2.00 -36.00
-3 1 -3.00 -90.00
-3 2 -1.50 -21.00
-6 1 -6.00 -552.00
1 1 1.00 -6.00
1 2 0.50 -6.00
2 1 2.00 0.00 x-2
3 1 3.00 24.00
3 2 1.50 -4.50
6 1 6.00 324.00
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
2x3-3x2+x-6
can be divided with x-2
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : 2x3-3x2+x-6
("Dividend")
By : x-2 ("Divisor")
dividend 2x3 - 3x2 + x - 6
- divisor * 2x2 2x3 - 4x2
remainder x2 + x - 6
- divisor * x1 x2 - 2x
remainder 3x - 6
- divisor * 3x0 3x - 6
remainder 0
Quotient : 2x2+x+3 Remainder: 0
Trying to factor by splitting the middle term
3.5 Factoring 2x2+x+3
The first term is, 2x2 its coefficient is 2 .
The middle term is, +x its coefficient is 1 .
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 1 .
-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
Multiplying Exponential Expressions:
3.6 Multiply (x-2) by (x-2)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (x-2) and the exponents are :
1 , as (x-2) is the same number as (x-2)1
and 1 , as (x-2) is the same number as (x-2)1
The product is therefore, (x-2)(1+1) = (x-2)2
Final result :
(2x2 + x + 3) • (x - 2)2