Rewrite using partial-fraction decomposition of \frac{(x^2+1)}{(x(x+1)^2)}

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

Question :

frac(x^2+1)(x(x+1)^2)

Simplify the expression

\frac{x ^{2} + 1}{x ^{3} + 2 x ^{2} + x}

Evaluate

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

Remove the parentheses

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

Solution

More Steps

Evaluate

x (x + 1)^{2}

Expand the expression

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Evaluate

(x + 1)^{2}

Use (a + b)^{2} = a^{2} + 2 ab + b^{2} to expand the expression

x^{2} + 2 x \times 1 + 1^{2}

Calculate

x^{2} + 2 x + 1

x (x^{2} + 2 x + 1)

Apply the distributive property

x \times x^{2} + x \times 2 x + x \times 1

Multiply the terms

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Evaluate

x \times x^{2}

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

x^{1 + 2}

Add the numbers

x^{3}

x^{3} + x \times 2 x + x \times 1

Multiply the terms

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Evaluate

x \times 2 x

Use the commutative property to reorder the terms

2 x \times x

Multiply the terms

2 x^{2}

x^{3} + 2 x^{2} + x \times 1

Any expression multiplied by 1 remains the same

x^{3} + 2 x^{2} + x

\frac{x ^{2} + 1}{x ^{3} + 2 x ^{2} + x}

Show Solution

Find the excluded values

x = 0, x = - 1

Evaluate

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

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

x (x + 1)^{2} = 0

Separate the equation into 2 possible cases

x = 0 (x + 1)^{2} = 0

Solve the equation

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Evaluate

(x + 1)^{2} = 0

The only way a power can be 0 is when the base equals 0

x + 1 = 0

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

x = 0 - 1

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

x = - 1

x = 0 x = - 1

Solution

x = 0, x = - 1

Show Solution

Rewrite the fraction

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

Evaluate

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

Remove the parentheses

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

For each factor in the denominator,write a new fraction

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

Write the terms in the numerator

\frac{A}{x} + \frac{B}{( x + 1 ) ^{2}} + \frac{C}{x + 1}

Set the sum of fractions equal to the original fraction

\frac{x ^{2} + 1}{x ( x + 1 ) ^{2}} = \frac{A}{x} + \frac{B}{( x + 1 ) ^{2}} + \frac{C}{x + 1}

Multiply both sides

\frac{x ^{2} + 1}{x ( x + 1 ) ^{2}} \times x (x + 1)^{2} = \frac{A}{x} \times x (x + 1)^{2} + \frac{B}{( x + 1 ) ^{2}} \times x (x + 1)^{2} + \frac{C}{x + 1} \times x (x + 1)^{2}

Simplify the expression

x^{2} + 1 = (x^{2} + 2 x + 1) A + x B + (x^{2} + x) C

Simplify the expression

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Evaluate

(x^{2} + 2 x + 1) A + x B + (x^{2} + x) C

Multiply the terms

A (x^{2} + 2 x + 1) + x B + (x^{2} + x) C

Multiply the terms

A (x^{2} + 2 x + 1) + x B + C (x^{2} + x)

Expand the expression

A x^{2} + 2 A x + A + x B + C (x^{2} + x)

Expand the expression

A x^{2} + 2 A x + A + x B + C x^{2} + C x

x^{2} + 1 = A x^{2} + 2 A x + A + x B + C x^{2} + C x

Group the terms

x^{2} + 1 = (A + C) x^{2} + (2 A + B + C) x + A

Equate the coefficients

⎩ ⎨ ⎧ 1 = A + C 0 = 2 A + B + C 1 = A

Swap the sides

⎩ ⎨ ⎧ A + C = 1 2 A + B + C = 0 A = 1

Substitute the given value of A into the equation {A + C = 1 2 A + B + C = 0

{1 + C = 1 2 \times 1 + B + C = 0

Any expression multiplied by 1 remains the same

{1 + C = 1 2 + B + C = 0

Solve the equation for C

More Steps

Evaluate

1 + C = 1

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

C = 1 - 1

Subtract the terms

C = 0

{C = 0 2 + B + C = 0

Substitute the given value of C into the equation 2 + B + C = 0

2 + B + 0 = 0

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

2 + B = 0

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

B = 0 - 2

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

B = - 2

Calculate

⎩ ⎨ ⎧ A = 1 B = - 2 C = 0

Solution

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

Show Solution

Find the roots

x \in / R

Evaluate

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

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

\frac{( x ^{2} + 1 )}{( x ( x + 1 ) ^{2} )} = 0

Find the domain

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Evaluate

x (x + 1)^{2} \neq = 0

Apply the zero product property

{x \neq = 0 (x + 1)^{2} \neq = 0

Solve the inequality

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Evaluate

(x + 1)^{2} \neq = 0

The only way a power can not be 0 is when the base not equals 0

x + 1 \neq = 0

Move the constant to the right side

x \neq = 0 - 1

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

x \neq = - 1

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

Find the intersection

x \in (- \infty, - 1) \cup (- 1, 0) \cup (0, + \infty)

\frac{( x ^{2} + 1 )}{( x ( x + 1 ) ^{2} )} = 0, x \in (- \infty, - 1) \cup (- 1, 0) \cup (0, + \infty)

Calculate

\frac{( x ^{2} + 1 )}{( x ( x + 1 ) ^{2} )} = 0

Remove the parentheses

\frac{x ^{2} + 1}{( x ( x + 1 ) ^{2} )} = 0

Multiply the terms

\frac{x ^{2} + 1}{x ( x + 1 ) ^{2}} = 0

Cross multiply

x^{2} + 1 = x (x + 1)^{2} \times 0

Simplify the equation

x^{2} + 1 = 0

Solution

x \in / R

Show Solution

frac(x^2+1)(x(x+1)^2)

Simplify the expression

Find the excluded values

Rewrite the fraction

Find the roots

Frequently Asked Questions

Solution of \(\frac{(x^2+1)}{(x(x+1)^2)}\)

Find the excluded values of \(\frac{(x^2+1)}{(x(x+1)^2)}\)

Rewrite using partial-fraction decomposition of \(\frac{(x^2+1)}{(x(x+1)^2)}\)

Find the roots of the algebra expression of \(\frac{(x^2+1)}{(x(x+1)^2)}\)