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Simplify the expression $\frac{xe^{\left(2x-6\right)}-7x+18}{x^3-5x^2+3x+9}$

Step-by-step Solution

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Final Answer

$\frac{xe^{\left(2x-6\right)}-7x+18}{\left(x-3\right)^2\left(x+1\right)}$
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Step-by-step Solution

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We can factor the polynomial $x^3-5x^2+3x+9$ 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 $9$

$1, 3, 9$

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$1, 3, 9$

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Learn how to solve simplification of algebraic fractions problems step by step online. Simplify the expression (xe^(2x-6)-7x+18)/(x^3-5x^23x+9). We can factor the polynomial x^3-5x^2+3x+9 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 9. Next, list all divisors of the leading coefficient a_n, which equals 1. The possible roots \pm\frac{p}{q} of the polynomial x^3-5x^2+3x+9 will then be. Trying all possible roots, we found that 3 is a root of the polynomial. When we evaluate it in the polynomial, it gives us 0 as a result.

Final Answer

$\frac{xe^{\left(2x-6\right)}-7x+18}{\left(x-3\right)^2\left(x+1\right)}$

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 (xe^(2x-6)+-7x)/(x^3+-5x^2) using the definitionSolve by quadratic formula (general formula)Find break even pointsFind the discriminant

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Function Plot

Plotting: $\frac{xe^{\left(2x-6\right)}-7x+18}{\left(x-3\right)^2\left(x+1\right)}$

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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: Simplification of algebraic fractions

Simplification or reduction of algebraic fractions is the action of dividing the numerator and denominator of a fraction by a common factor in order to obtain another much simpler equivalent fraction. We can say that a fraction is reduced to its simplest when there is no common factor between the numerator and the denominator.

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