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Simplify the expression $\frac{x^{10}+2x^6+x^2-1}{x^3-x^2+1}$

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Final Answer

$x^{7}+x^{6}+x^{5}+x^{3}-1$
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Step-by-step Solution

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1

Divide $x^{10}+2x^6+x^2-1$ by $x^3-x^2+1$

$\begin{array}{l}\phantom{\phantom{;}x^{3}-x^{2}+1;}{\phantom{;}x^{7}+x^{6}+x^{5}\phantom{-;x^n}+x^{3}\phantom{-;x^n}\phantom{-;x^n}-1\phantom{;}\phantom{;}}\\\phantom{;}x^{3}-x^{2}+1\overline{\smash{)}\phantom{;}x^{10}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+2x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{3}-x^{2}+1;}\underline{-x^{10}+x^{9}\phantom{-;x^n}-x^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{10}+x^{9}-x^{7};}\phantom{;}x^{9}\phantom{-;x^n}-x^{7}+2x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}-x^{2}+1-;x^n;}\underline{-x^{9}+x^{8}\phantom{-;x^n}-x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-x^{9}+x^{8}-x^{6}-;x^n;}\phantom{;}x^{8}-x^{7}+x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}-x^{2}+1-;x^n-;x^n;}\underline{-x^{8}+x^{7}\phantom{-;x^n}-x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-x^{8}+x^{7}-x^{5}-;x^n-;x^n;}\phantom{;}x^{6}-x^{5}\phantom{-;x^n}\phantom{-;x^n}+x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}-x^{2}+1-;x^n-;x^n-;x^n;}\underline{-x^{6}+x^{5}\phantom{-;x^n}-x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;-x^{6}+x^{5}-x^{3}-;x^n-;x^n-;x^n;}-x^{3}+x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}-x^{2}+1-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}x^{3}-x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{;;;;\phantom{;}x^{3}-x^{2}+1\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}\\\end{array}$
2

Resulting polynomial

$x^{7}+x^{6}+x^{5}+x^{3}-1$

Final Answer

$x^{7}+x^{6}+x^{5}+x^{3}-1$

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 (x^10+2x^6)/(x^3+-1x^2) using the definitionSolve by quadratic formula (general formula)Find the rootsFind break even pointsFind the discriminant

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Plotting: $x^{7}+x^{6}+x^{5}+x^{3}-1$

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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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