Answer:
step by step
STEP
1
:
Equation at the end of step 1
((((3•(m3))+5m2)-5m)+1)
(———————————————————————•m)-1
3
STEP
2
:
Equation at the end of step
2
:
(((3m3+5m2)-5m)+1)
(——————————————————•m)-1
3
STEP
3
:
3m3 + 5m2 - 5m + 1
Simplify ——————————————————
3
Checking for a perfect cube :
3.1 3m3 + 5m2 - 5m + 1 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 3m3 + 5m2 - 5m + 1
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -5m + 1
Group 2: 5m2 + 3m3
Pull out from each group separately :
Group 1: (-5m + 1) • (1) = (5m - 1) • (-1)
Group 2: (3m + 5) • (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 :
3.3 Find roots (zeroes) of : F(m) = 3m3 + 5m2 - 5m + 1
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 3 and the Trailing Constant is 1.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 8.00
-1 3 -0.33 3.11
1 1 1.00 4.00
1 3 0.33 0.00 3m - 1
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
3m3 + 5m2 - 5m + 1
can be divided with 3m - 1
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : 3m3 + 5m2 - 5m + 1
("Dividend")
By : 3m - 1 ("Divisor")
dividend 3m3 + 5m2 - 5m + 1
- divisor * m2 3m3 - m2
remainder 6m2 - 5m + 1
- divisor * 2m1 6m2 - 2m
remainder - 3m + 1
- divisor * -m0 - 3m + 1
remainder 0
Quotient : m2+2m-1 Remainder: 0
Trying to factor by splitting the middle term
3.5 Factoring m2+2m-1
The first term is, m2 its coefficient is 1 .
The middle term is, +2m its coefficient is 2 .
The last term, "the constant", is -1
Step-1 : Multiply the coefficient of the first term by the constant 1 • -1 = -1
Step-2 : Find two factors of -1 whose sum equals the coefficient of the middle term, which is 2 .
-1 + 1 = 0
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step
3
:
(m2 + 2m - 1) • (3m - 1)
(———————————————————————— • m) - 1
3
STEP
4
:
Equation at the end of step 4
m • (m2 + 2m - 1) • (3m - 1)
———————————————————————————— - 1
3
STEP
5
:
Rewriting the whole as an Equivalent Fraction
5.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using 3 as the denominator :
1 1 • 3
1 = — = —————
1 3
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
5.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
m • (m2+2m-1) • (3m-1) - (3) 3m4 + 5m3 - 5m2 + m - 3
———————————————————————————— = ———————————————————————
3 3
Polynomial Roots Calculator :
5.3 Find roots (zeroes) of : F(m) = 3m4 + 5m3 - 5m2 + m - 3
See theory in step 3.3
In this case, the Leading Coefficient is 3 and the Trailing Constant is -3.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,3
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -11.00
-1 3 -0.33 -4.04
-3 1 -3.00 57.00
1 1 1.00 1.00
1 3 0.33 -3.00
3 1 3.00 333.00
Polynomial Roots Calculator found no rational roots
Final result :
3m4 + 5m3 - 5m2 + m - 3
———————————————————————
3
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