m3 by (4m2 – 3m + 8 )
Answers
Answer:
Final result :
(m2 - 2m + 2) • (m - 1)
Step-by-step explanation:
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "m2" was replaced by "m^2". 1 more similar replacement(s).
STEP
1
:
Equation at the end of step 1
(((m3) - 3m2) + 4m) - 2
STEP
2
:
Checking for a perfect cube
2.1 m3-3m2+4m-2 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: m3-3m2+4m-2
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 4m-2
Group 2: m3-3m2
Pull out from each group separately :
Group 1: (2m-1) • (2)
Group 2: (m-3) • (m2)
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(m) = m3-3m2+4m-2
Polynomial Roots Calculator is a set of methods aimed at finding values of m for which F(m)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers m 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 1 and the Trailing Constant is -2.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -10.00
-2 1 -2.00 -30.00
1 1 1.00 0.00 m-1
2 1 2.00 2.00
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
m3-3m2+4m-2
can be divided with m-1
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : m3-3m2+4m-2
("Dividend")
By : m-1 ("Divisor")
dividend m3 - 3m2 + 4m - 2
- divisor * m2 m3 - m2
remainder - 2m2 + 4m - 2
- divisor * -2m1 - 2m2 + 2m
remainder 2m - 2
- divisor * 2m0 2m - 2
remainder 0
Quotient : m2-2m+2 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring m2-2m+2
The first term is, m2 its coefficient is 1 .
The middle term is, -2m its coefficient is -2 .
The last term, "the constant", is +2
Step-1 : Multiply the coefficient of the first term by the constant 1 • 2 = 2
Step-2 : Find two factors of 2 whose sum equals the coefficient of the middle term, which is -2 .
-2 + -1 = -3
-1 + -2 = -3
1 + 2 = 3
2 + 1 = 3