What is the remainder when (m² - 8m + 12 ) is divided by (m+ 6)?
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Reducing fractions to their lowest terms
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We think you wrote:
m2-8m+12:m+16
This deals with reducing fractions to their lowest terms.
Overview
Steps
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1 solution(s) found
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Step by Step Solution
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Reformatting the input :
Changes made to your input should not affect the solution:
(1): "m2" was replaced by "m^2".
STEP
1
:
12
Simplify ——
m
Equation at the end of step
1
:
12
(((m2) - 8m) + ——) + 16
m
STEP
2
:
Rewriting the whole as an Equivalent Fraction
2.1 Adding a fraction to a whole
Rewrite the whole as a fraction using m as the denominator :
m2 - 8m (m2 - 8m) • m
m2 - 8m = ——————— = —————————————
1 m
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
3
:
Pulling out like terms
3.1 Pull out like factors :
m2 - 8m = m • (m - 8)
Adding fractions that have a common denominator :
3.2 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 • (m-8) • m + 12 m3 - 8m2 + 12
—————————————————— = —————————————
m m
Equation at the end of step
3
:
(m3 - 8m2 + 12)
——————————————— + 16
m
STEP
4
:
Rewriting the whole as an Equivalent Fraction :
4.1 Adding a whole to a fraction
Rewrite the whole as a fraction using m as the denominator :
16 16 • m
16 = —— = ——————
1 m
Polynomial Roots Calculator :
4.2 Find roots (zeroes) of : F(m) = m3 - 8m2 + 12
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 12.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,12
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 3.00
-2 1 -2.00 -28.00
-3 1 -3.00 -87.00
-4 1 -4.00 -180.00
-6 1 -6.00 -492.00
-12 1 -12.00 -2868.00
1 1 1.00 5.00
2 1 2.00 -12.00
3 1 3.00 -33.00
4 1 4.00 -52.00
6 1 6.00 -60.00
12 1 12.00 588.00
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
4.3 Adding up the two equivalent fractions
(m3-8m2+12) + 16 • m m3 - 8m2 + 16m + 12
———————————————————— = ———————————————————
m m
Checking for a perfect cube :
4.4 m3 - 8m2 + 16m + 12 is not a perfect cube
Trying to factor by pulling out :
4.5 Factoring: m3 - 8m2 + 16m + 12
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: m3 + 12
Group 2: -8m2 + 16m
Pull out from each group separately :
Group 1: (m3 + 12) • (1)
Group 2: (m - 2) • (-8m)
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 :
4.6 Find roots (zeroes) of : F(m) = m3 - 8m2 + 16m + 12
See theory in step 4.2
In this case, the Leading Coefficient is 1 and the Trailing Constant is 12.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,12
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -13.00
-2 1 -2.00 -60.00
-3 1 -3.00 -135.00
-4 1 -4.00 -244.00
-6 1 -6.00 -588.00
-12 1 -12.00 -3060.00
1 1 1.00 21.00
2 1 2.00 20.00
3 1 3.00 15.00
4 1 4.00 12.00
6 1 6.00 36.00
12 1 12.00 780.00
Polynomial Roots Calculator found no rational roots
Final result :
m3 - 8m2 + 16m + 12
———————————————————
m
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