Solve ( x^4 + x^3 - x^2 + 2 x ) / ( x^2 + 2 x )

Question

Simplify the expression

x^{2} - x + 1

Evaluate

\frac{x ^{4} + x ^{3} - x ^{2} + 2 x}{x ^{2} + 2 x}

Factor the expression

\frac{( x ^{2} + 2 x ) ( x ^{2} - x + 1 )}{x ^{2} + 2 x}

Solution

x^{2} - x + 1

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Find the excluded values

x = 0, x = - 2

Evaluate

\frac{x ^{4} + x ^{3} - x ^{2} + 2 x}{x ^{2} + 2 x}

To find the excluded values,set the denominators equal to 0

x^{2} + 2 x = 0

Factor the expression

More Steps

Evaluate

x^{2} + 2 x

Rewrite the expression

x \times x + x \times 2

Factor out x from the expression

x (x + 2)

x (x + 2) = 0

When the product of factors equals 0,at least one factor is 0

x = 0 x + 2 = 0

Solve the equation for x

More Steps

Evaluate

x + 2 = 0

Move the constant to the right-hand side and change its sign

x = 0 - 2

Removing 0 doesn't change the value,so remove it from the expression

x = - 2

x = 0 x = - 2

Solution

x = 0, x = - 2

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Find the roots

x \in / R

Evaluate

\frac{x ^{4} + x ^{3} - x ^{2} + 2 x}{x ^{2} + 2 x}

To find the roots of the expression,set the expression equal to 0

\frac{x ^{4} + x ^{3} - x ^{2} + 2 x}{x ^{2} + 2 x} = 0

Find the domain

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Evaluate

x^{2} + 2 x \neq = 0

Add the same value to both sides

x^{2} + 2 x + 1 \neq = 1

Evaluate

(x + 1)^{2} \neq = 1

Take the root of both sides of the equation and remember to use both positive and negative roots

x + 1 \neq = \pm 1

Simplify the expression

x + 1 \neq = \pm 1

Separate the inequality into 2 possible cases

{x + 1 \neq = 1 x + 1 \neq = - 1

Cancel equal terms on both sides of the expression

{x \neq = 0 x + 1 \neq = - 1

Calculate

More Steps

Evaluate

x + 1 \neq = - 1

Move the constant to the right side

x \neq = - 1 - 1

Subtract the numbers

x \neq = - 2

{x \neq = 0 x \neq = - 2

Find the intersection

x \in (- \infty, - 2) \cup (- 2, 0) \cup (0, + \infty)

\frac{x ^{4} + x ^{3} - x ^{2} + 2 x}{x ^{2} + 2 x} = 0, x \in (- \infty, - 2) \cup (- 2, 0) \cup (0, + \infty)

Calculate

\frac{x ^{4} + x ^{3} - x ^{2} + 2 x}{x ^{2} + 2 x} = 0

Divide the terms

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Evaluate

\frac{x ^{4} + x ^{3} - x ^{2} + 2 x}{x ^{2} + 2 x}

Factor the expression

\frac{( x ^{2} + 2 x ) ( x ^{2} - x + 1 )}{x ^{2} + 2 x}

Reduce the fraction

x^{2} - x + 1

x^{2} - x + 1 = 0

Substitute a= 1,b= - 1 and c= 1 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{1 \pm ( - 1 ) ^{2} - 4}{2}

Simplify the expression

More Steps

Evaluate

(- 1)^{2} - 4

Evaluate the power

1 - 4

Subtract the numbers

- 3

x = \frac{1 \pm - 3}{2}

Solution

x \in / R

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