Simplify the expression $g\left(x\right)=\frac{x^3+5x}{x^5+3x^3-4x}$

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Final answer to the problem

$g\left(x\right)=\frac{x^2+5}{\left(x^2+4\right)\left(x^2-1\right)}$
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We can factor the polynomial $x^5+3x^3-4x$ using the rational root theorem, which guarantees that for a polynomial of the form $a_nx^n+a_{n-1}x^{n-1}+\dots+a_0$ there is a rational root of the form $\pm\frac{p}{q}$, where $p$ belongs to the divisors of the constant term $a_0$, and $q$ belongs to the divisors of the leading coefficient $a_n$. List all divisors $p$ of the constant term $a_0$, which equals $0$

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Learn how to solve algebraic expressions problems step by step online. Simplify the expression g(x)=(x^3+5x)/(x^5+3x^3-4x). We can factor the polynomial x^5+3x^3-4x using the rational root theorem, which guarantees that for a polynomial of the form a_nx^n+a_{n-1}x^{n-1}+\dots+a_0 there is a rational root of the form \pm\frac{p}{q}, where p belongs to the divisors of the constant term a_0, and q belongs to the divisors of the leading coefficient a_n. List all divisors p of the constant term a_0, which equals 0. Next, list all divisors of the leading coefficient a_n, which equals 1. The possible roots \pm\frac{p}{q} of the polynomial x^5+3x^3-4x will then be. We can factor the polynomial x^5+3x^3-4x using synthetic division (Ruffini's rule). We found that 1 is a root of the polynomial.

Final answer to the problem

$g\left(x\right)=\frac{x^2+5}{\left(x^2+4\right)\left(x^2-1\right)}$

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Function Plot

Plotting: $g\left(x\right)+\frac{-x^3-5x}{x^5+3x^3-4x}$

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1
2
3
4
5
6
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

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Main Topic: Algebraic expressions

An algebraic expression is a group of terms that are separated by $+$ or $-$ signs.

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