12m^(5)-9m^(3)+16-6m^(2)+8m
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Reducing fractions to their lowest terms
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We think you wrote:
12m^4-24m^3+6m^2+9m-9/6m^2
This deals with reducing fractions to their lowest terms.
Overview
Steps
Topics
1 result(s) found
2
3m⋅(8m
3
−16m
2
+3m+6)
See steps
Step by Step Solution:
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STEP
1
:
3
Simplify —
2
Equation at the end of step
1
:
3
((((12•(m4))-(24•(m3)))+(6•(m2)))+9m)-(—•m2)
2
STEP
2
:
Equation at the end of step 2
3m2
((((12•(m4))-(24•(m3)))+(6•(m2)))+9m)-———
2
STEP
3
:
Equation at the end of step
3
:
3m2
((((12•(m4))-(24•(m3)))+(2•3m2))+9m)-———
2
STEP
4
:
Equation at the end of step
4
:
3m2
((((12•(m4))-(23•3m3))+(2•3m2))+9m)-———
2
STEP
5
:
Equation at the end of step
5
:
3m2
((((22•3m4) - (23•3m3)) + (2•3m2)) + 9m) - ———
2
STEP
6
:
Rewriting the whole as an Equivalent Fraction
6.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using 2 as the denominator :
12m4 - 24m3 + 6m2 + 9m (12m4 - 24m3 + 6m2 + 9m) • 2
12m4 - 24m3 + 6m2 + 9m = —————————————————————— = ————————————————————————————
1 2
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
STEP
7
:
Pulling out like terms
7.1 Pull out like factors :
12m4 - 24m3 + 6m2 + 9m =
3m • (4m3 - 8m2 + 2m + 3)
Checking for a perfect cube :
7.2 4m3 - 8m2 + 2m + 3 is not a perfect cube
Trying to factor by pulling out :
7.3 Factoring: 4m3 - 8m2 + 2m + 3
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 2m + 3
Group 2: 4m3 - 8m2
Pull out from each group separately :
Group 1: (2m + 3) • (1)
Group 2: (m - 2) • (4m2)
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 :
7.4 Find roots (zeroes) of : F(m) = 4m3 - 8m2 + 2m + 3
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 4 and the Trailing Constant is 3.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
of the Trailing Constant : 1 ,3
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -11.00
-1 2 -0.50 -0.50
-1 4 -0.25 1.94
-3 1 -3.00 -183.00
-3 2 -1.50 -31.50
-3 4 -0.75 -4.69
1 1 1.00 1.00
1 2 0.50 2.50
1 4 0.25 3.06
3 1 3.00 45.00
3 2 1.50 1.50
3 4 0.75 1.69
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
7.5 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:
3m • (4m3-8m2+2m+3) • 2 - (3m2) 24m4 - 48m3 + 9m2 + 18m
——————————————————————————————— = ———————————————————————
2 2
STEP
8
:
Pulling out like terms :
8.1 Pull out like factors :
24m4 - 48m3 + 9m2 + 18m =
3m • (8m3 - 16m2 + 3m + 6)
Checking for a perfect cube :
8.2 8m3 - 16m2 + 3m + 6 is not a perfect cube
Trying to factor by pulling out :
8.3 Factoring: 8m3 - 16m2 + 3m + 6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 3m + 6
Group 2: 8m3 - 16m2
Pull out from each group separately :
Group 1: (m + 2) • (3)
Group 2: (m - 2) • (8m2)
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 :
8.4 Find roots (zeroes) of : F(m) = 8m3 - 16m2 + 3m + 6
See theory in step 7.4
In this case, the Leading Coefficient is 8 and the Trailing Constant is 6.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4 ,8
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -21.00
-1 2 -0.50 -0.50
-1 4 -0.25 4.12
-1 8 -0.12 5.36
-2 1 -2.00 -128.00
Note - For tidiness, printing of 15 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Final result :
3m • (8m3 - 16m2 + 3m + 6)
——————————————————————————
2
Terms and topics
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Dividing exponents
Polynomial root calculator
NonLinear equations
Equations which are reducible to quadratic
Reducing fractions to lowest terms