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Simplify the expression $\frac{12m^2+19m^4-11m^3+3m+30m^5+1}{4m^2+5m^3-2m+1}$

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Final Answer

$6m^{2}-m+1+\frac{6m}{4m^2+5m^3-2m+1}$
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Step-by-step Solution

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1

Divide $12m^2+19m^4-11m^3+3m+30m^5+1$ by $4m^2+5m^3-2m+1$

$\begin{array}{l}\phantom{\phantom{;}5m^{3}+4m^{2}-2m\phantom{;}+1;}{\phantom{;}6m^{2}-m\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{;}5m^{3}+4m^{2}-2m\phantom{;}+1\overline{\smash{)}\phantom{;}30m^{5}+19m^{4}-11m^{3}+12m^{2}+3m\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}5m^{3}+4m^{2}-2m\phantom{;}+1;}\underline{-30m^{5}-24m^{4}+12m^{3}-6m^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-30m^{5}-24m^{4}+12m^{3}-6m^{2};}-5m^{4}+m^{3}+6m^{2}+3m\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}5m^{3}+4m^{2}-2m\phantom{;}+1-;x^n;}\underline{\phantom{;}5m^{4}+4m^{3}-2m^{2}+m\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}5m^{4}+4m^{3}-2m^{2}+m\phantom{;}-;x^n;}\phantom{;}5m^{3}+4m^{2}+4m\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}5m^{3}+4m^{2}-2m\phantom{;}+1-;x^n-;x^n;}\underline{-5m^{3}-4m^{2}+2m\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{;;-5m^{3}-4m^{2}+2m\phantom{;}-1\phantom{;}\phantom{;}-;x^n-;x^n;}\phantom{;}6m\phantom{;}\phantom{-;x^n}\\\end{array}$
2

Resulting polynomial

$6m^{2}-m+1+\frac{6m}{4m^2+5m^3-2m+1}$

Final Answer

$6m^{2}-m+1+\frac{6m}{4m^2+5m^3-2m+1}$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

SimplifyWrite in simplest formFactorFactor by completing the squareFind the integralFind the derivativeFind (12m^2+19m^4)/(4m^2+5m^3) using the definitionSolve by quadratic formula (general formula)Find the rootsFind break even pointsFind the discriminant

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Function Plot

Plotting: $6m^{2}-m+1+\frac{6m}{4m^2+5m^3-2m+1}$

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5
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7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

How to improve your answer:

Main Topic: Polynomial long division

In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division.

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