Precalculus Examples

$\frac{3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3}{x^{5} + 2 x^{3} + x}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Factor the fraction.

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Step 1.1.1

Factor $x$ out of $x^{5} + 2 x^{3} + x$ .

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Step 1.1.1.1

Factor $x$ out of $x^{5}$ .

$\frac{3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3}{x \cdot x^{4} + 2 x^{3} + x}$

Step 1.1.1.2

Factor $x$ out of $2 x^{3}$ .

$\frac{3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3}{x \cdot x^{4} + x (2 x^{2}) + x}$

Step 1.1.1.3

Raise $x$ to the power of $1$ .

$\frac{3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3}{x \cdot x^{4} + x (2 x^{2}) + x^{1}}$

Step 1.1.1.4

Factor $x$ out of $x^{1}$ .

$\frac{3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3}{x \cdot x^{4} + x (2 x^{2}) + x \cdot 1}$

Step 1.1.1.5

Factor $x$ out of $x \cdot x^{4} + x (2 x^{2})$ .

$\frac{3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3}{x (x^{4} + 2 x^{2}) + x \cdot 1}$

Step 1.1.1.6

Factor $x$ out of $x (x^{4} + 2 x^{2}) + x \cdot 1$ .

$\frac{3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3}{x (x^{4} + 2 x^{2} + 1)}$

Step 1.1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3}{x ({(x^{2})}^{2} + 2 x^{2} + 1)}$

Step 1.1.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$\frac{3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3}{x (u^{2} + 2 u + 1)}$

Step 1.1.4

Factor using the perfect square rule.

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Step 1.1.4.1

Rewrite $1$ as $1^{2}$ .

$\frac{3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3}{x (u^{2} + 2 u + 1^{2})}$

Step 1.1.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 u = 2 \cdot u \cdot 1$

Step 1.1.4.3

Rewrite the polynomial.

$\frac{3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3}{x (u^{2} + 2 \cdot u \cdot 1 + 1^{2})}$

Step 1.1.4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 1$ .

$\frac{3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3}{x {(u + 1)}^{2}}$

Step 1.1.5

Replace all occurrences of $u$ with $x^{2}$ .

$\frac{3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3}{x {(x^{2} + 1)}^{2}}$

Step 1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor is 2nd order, $2$ terms are required in the numerator. The number of terms required in the numerator is always equal to the order of the factor in the denominator.

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

Step 1.3

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

Step 1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $x {(x^{2} + 1)}^{2}$ .

$\frac{(3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3) (x {(x^{2} + 1)}^{2})}{x {(x^{2} + 1)}^{2}} = \frac{A (x {(x^{2} + 1)}^{2})}{x} + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.5

Cancel the common factor of $x$ .

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Step 1.5.1

Cancel the common factor.

Step 1.5.2

Rewrite the expression.

$\frac{(3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3) ({(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} = \frac{A (x {(x^{2} + 1)}^{2})}{x} + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.6

Cancel the common factor of ${(x^{2} + 1)}^{2}$ .

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Step 1.6.1

Cancel the common factor.

$\frac{(3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3) {(x^{2} + 1)}^{2}}{{(x^{2} + 1)}^{2}} = \frac{A (x {(x^{2} + 1)}^{2})}{x} + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.6.2

Divide $3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3$ by $1$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = \frac{A (x {(x^{2} + 1)}^{2})}{x} + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7

Simplify each term.

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Step 1.7.1

Cancel the common factor of $x$ .

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Step 1.7.1.1

Cancel the common factor.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = \frac{A (x {(x^{2} + 1)}^{2})}{x} + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.1.2

Divide $A ({(x^{2} + 1)}^{2})$ by $1$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A ({(x^{2} + 1)}^{2}) + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.2

Rewrite ${(x^{2} + 1)}^{2}$ as $(x^{2} + 1) (x^{2} + 1)$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A ((x^{2} + 1) (x^{2} + 1)) + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.3

Expand $(x^{2} + 1) (x^{2} + 1)$ using the FOIL Method.

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Step 1.7.3.1

Apply the distributive property.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A (x^{2} (x^{2} + 1) + 1 (x^{2} + 1)) + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.3.2

Apply the distributive property.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A (x^{2} x^{2} + x^{2} \cdot 1 + 1 (x^{2} + 1)) + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.3.3

Apply the distributive property.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A (x^{2} x^{2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1) + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.4

Simplify and combine like terms.

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Step 1.7.4.1

Simplify each term.

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Step 1.7.4.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.7.4.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A (x^{2 + 2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1) + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.4.1.1.2

Add $2$ and $2$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A (x^{4} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1) + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.4.1.2

Multiply $x^{2}$ by $1$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A (x^{4} + x^{2} + 1 x^{2} + 1 \cdot 1) + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.4.1.3

Multiply $x^{2}$ by $1$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A (x^{4} + x^{2} + x^{2} + 1 \cdot 1) + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.4.1.4

Multiply $1$ by $1$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A (x^{4} + x^{2} + x^{2} + 1) + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.4.2

Add $x^{2}$ and $x^{2}$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A (x^{4} + 2 x^{2} + 1) + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.5

Apply the distributive property.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + A (2 x^{2}) + A \cdot 1 + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.6

Simplify.

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Step 1.7.6.1

Rewrite using the commutative property of multiplication.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A \cdot 1 + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.6.2

Multiply $A$ by $1$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.7

Cancel the common factor of ${(x^{2} + 1)}^{2}$ .

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Step 1.7.7.1

Cancel the common factor.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + \frac{(B x + C) (x {(x^{2} + 1)}^{2})}{{(x^{2} + 1)}^{2}} + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.7.2

Divide $(B x + C) (x)$ by $1$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + (B x + C) (x) + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.8

Apply the distributive property.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x \cdot x + C x + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.9

Multiply $x$ by $x$ by adding the exponents.

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Step 1.7.9.1

Move $x$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B (x \cdot x) + C x + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.9.2

Multiply $x$ by $x$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + \frac{(D x + F) (x {(x^{2} + 1)}^{2})}{x^{2} + 1}$

Step 1.7.10

Cancel the common factor of ${(x^{2} + 1)}^{2}$ and $x^{2} + 1$ .

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Step 1.7.10.1

Factor $x^{2} + 1$ out of $(D x + F) (x {(x^{2} + 1)}^{2})$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + \frac{(x^{2} + 1) ((D x + F) (x (x^{2} + 1)))}{x^{2} + 1}$

Step 1.7.10.2

Cancel the common factors.

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Step 1.7.10.2.1

Multiply by $1$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + \frac{(x^{2} + 1) ((D x + F) (x (x^{2} + 1)))}{(x^{2} + 1) \cdot 1}$

Step 1.7.10.2.2

Cancel the common factor.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + \frac{(x^{2} + 1) ((D x + F) (x (x^{2} + 1)))}{(x^{2} + 1) \cdot 1}$

Step 1.7.10.2.3

Rewrite the expression.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + \frac{(D x + F) (x (x^{2} + 1))}{1}$

Step 1.7.10.2.4

Divide $(D x + F) (x (x^{2} + 1))$ by $1$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + (D x + F) (x (x^{2} + 1))$

Step 1.7.11

Apply the distributive property.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + (D x + F) (x \cdot x^{2} + x \cdot 1)$

Step 1.7.12

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.7.12.1

Multiply $x$ by $x^{2}$ .

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Step 1.7.12.1.1

Raise $x$ to the power of $1$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + (D x + F) (x \cdot x^{2} + x \cdot 1)$

Step 1.7.12.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + (D x + F) (x^{1 + 2} + x \cdot 1)$

Step 1.7.12.2

Add $1$ and $2$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + (D x + F) (x^{3} + x \cdot 1)$

Step 1.7.13

Multiply $x$ by $1$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + (D x + F) (x^{3} + x)$

Step 1.7.14

Expand $(D x + F) (x^{3} + x)$ using the FOIL Method.

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Step 1.7.14.1

Apply the distributive property.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + D x (x^{3} + x) + F (x^{3} + x)$

Step 1.7.14.2

Apply the distributive property.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + D x \cdot x^{3} + D x \cdot x + F (x^{3} + x)$

Step 1.7.14.3

Apply the distributive property.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + D x \cdot x^{3} + D x \cdot x + F x^{3} + F x$

Step 1.7.15

Simplify each term.

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Step 1.7.15.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 1.7.15.1.1

Move $x^{3}$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + D (x^{3} x) + D x \cdot x + F x^{3} + F x$

Step 1.7.15.1.2

Multiply $x^{3}$ by $x$ .

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Step 1.7.15.1.2.1

Raise $x$ to the power of $1$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + D (x^{3} x) + D x \cdot x + F x^{3} + F x$

Step 1.7.15.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + D x^{3 + 1} + D x \cdot x + F x^{3} + F x$

Step 1.7.15.1.3

Add $3$ and $1$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + D x^{4} + D x \cdot x + F x^{3} + F x$

Step 1.7.15.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.7.15.2.1

Move $x$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + D x^{4} + D (x \cdot x) + F x^{3} + F x$

Step 1.7.15.2.2

Multiply $x$ by $x$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 A x^{2} + A + B x^{2} + C x + D x^{4} + D x^{2} + F x^{3} + F x$

Step 1.8

Simplify the expression.

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Step 1.8.1

Move $A$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 x^{2} A + A + B x^{2} + C x + D x^{4} + D x^{2} + F x^{3} + F x$

Step 1.8.2

Reorder $B$ and $x^{2}$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 x^{2} A + A + B x^{2} + C x + D x^{4} + D x^{2} + F x^{3} + F x$

Step 1.8.3

Reorder $F$ and $x^{3}$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 x^{2} A + A + B x^{2} + C x + D x^{4} + D x^{2} + F x^{3} + F x$

Step 1.8.4

Reorder $F$ and $x$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 x^{2} A + A + B x^{2} + C x + D x^{4} + D x^{2} + F x^{3} + F x$

Step 1.8.5

Move $A$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 x^{2} A + B x^{2} + C x + D x^{4} + D x^{2} + F x^{3} + F x + A$

Step 1.8.6

Move $C x$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 x^{2} A + B x^{2} + D x^{4} + D x^{2} + F x^{3} + C x + F x + A$

Step 1.8.7

Move $D x^{2}$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 x^{2} A + B x^{2} + D x^{4} + F x^{3} + D x^{2} + C x + F x + A$

Step 1.8.8

Move $B x^{2}$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + 2 x^{2} A + D x^{4} + F x^{3} + B x^{2} + D x^{2} + C x + F x + A$

Step 1.8.9

Move $2 x^{2} A$ .

$3 x^{4} + x^{3} + 4 x^{2} + 6 x + 3 = A x^{4} + D x^{4} + F x^{3} + 2 x^{2} A + B x^{2} + D x^{2} + C x + F x + A$

Step 2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 2.1

Create an equation for the partial fraction variables by equating the coefficients of $x^{4}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$3 = A + D$

Step 2.2

Create an equation for the partial fraction variables by equating the coefficients of $x^{3}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = F$

Step 2.3

Create an equation for the partial fraction variables by equating the coefficients of $x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$4 = 2 A + B + D$

Step 2.4

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$6 = C + F$

Step 2.5

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$3 = A$

Step 2.6

Set up the system of equations to find the coefficients of the partial fractions.

$3 = A + D$

$1 = F$

$4 = 2 A + B + D$

$6 = C + F$

$3 = A$

$3 = A + D$

$1 = F$

$4 = 2 A + B + D$

$6 = C + F$

$3 = A$

Step 3

Solve the system of equations.

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Step 3.1

Rewrite the equation as $F = 1$ .

$F = 1$

$3 = A + D$

$4 = 2 A + B + D$

$6 = C + F$

$3 = A$

Step 3.2

Replace all occurrences of $F$ with $1$ in each equation.

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Step 3.2.1

Replace all occurrences of $F$ in $6 = C + F$ with $1$ .

$6 = C + 1$

$F = 1$

$3 = A + D$

$4 = 2 A + B + D$

$3 = A$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Remove parentheses.

$6 = C + 1$

$F = 1$

$3 = A + D$

$4 = 2 A + B + D$

$3 = A$

$6 = C + 1$

$F = 1$

$3 = A + D$

$4 = 2 A + B + D$

$3 = A$

Step 3.2.3

Rewrite the equation as $A = 3$ .

$A = 3$

$6 = C + 1$

$F = 1$

$3 = A + D$

$4 = 2 A + B + D$

$A = 3$

$6 = C + 1$

$F = 1$

$3 = A + D$

$4 = 2 A + B + D$

Step 3.3

Replace all occurrences of $A$ with $3$ in each equation.

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Step 3.3.1

Rewrite the equation as $C + 1 = 6$ .

$C + 1 = 6$

$A = 3$

$F = 1$

$3 = A + D$

$4 = 2 A + B + D$

Step 3.3.2

Move all terms not containing $C$ to the right side of the equation.

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Step 3.3.2.1

Subtract $1$ from both sides of the equation.

$C = 6 - 1$

$A = 3$

$F = 1$

$3 = A + D$

$4 = 2 A + B + D$

Step 3.3.2.2

Subtract $1$ from $6$ .

$C = 5$

$A = 3$

$F = 1$

$3 = A + D$

$4 = 2 A + B + D$

$C = 5$

$A = 3$

$F = 1$

$3 = A + D$

$4 = 2 A + B + D$

Step 3.3.3

Replace all occurrences of $A$ in $3 = A + D$ with $3$ .

$3 = (3) + D$

$C = 5$

$A = 3$

$F = 1$

$4 = 2 A + B + D$

Step 3.3.4

Simplify the right side.

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Step 3.3.4.1

Remove parentheses.

$3 = 3 + D$

$C = 5$

$A = 3$

$F = 1$

$4 = 2 A + B + D$

$3 = 3 + D$

$C = 5$

$A = 3$

$F = 1$

$4 = 2 A + B + D$

Step 3.3.5

Replace all occurrences of $A$ in $4 = 2 A + B + D$ with $3$ .

$4 = 2 (3) + B + D$

$3 = 3 + D$

$C = 5$

$A = 3$

$F = 1$

Step 3.3.6

Simplify the right side.

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Step 3.3.6.1

Multiply $2$ by $3$ .

$4 = 6 + B + D$

$3 = 3 + D$

$C = 5$

$A = 3$

$F = 1$

$4 = 6 + B + D$

$3 = 3 + D$

$C = 5$

$A = 3$

$F = 1$

$4 = 6 + B + D$

$3 = 3 + D$

$C = 5$

$A = 3$

$F = 1$

Step 3.4

Solve for $D$ in $3 = 3 + D$ .

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Step 3.4.1

Rewrite the equation as $3 + D = 3$ .

$3 + D = 3$

$4 = 6 + B + D$

$C = 5$

$A = 3$

$F = 1$

Step 3.4.2

Move all terms not containing $D$ to the right side of the equation.

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Step 3.4.2.1

Subtract $3$ from both sides of the equation.

$D = 3 - 3$

$4 = 6 + B + D$

$C = 5$

$A = 3$

$F = 1$

Step 3.4.2.2

Subtract $3$ from $3$ .

$D = 0$

$4 = 6 + B + D$

$C = 5$

$A = 3$

$F = 1$

$D = 0$

$4 = 6 + B + D$

$C = 5$

$A = 3$

$F = 1$

$D = 0$

$4 = 6 + B + D$

$C = 5$

$A = 3$

$F = 1$

Step 3.5

Replace all occurrences of $D$ with $0$ in each equation.

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Step 3.5.1

Replace all occurrences of $D$ in $4 = 6 + B + D$ with $0$ .

$4 = 6 + B + 0$

$D = 0$

$C = 5$

$A = 3$

$F = 1$

Step 3.5.2

Simplify $4 = 6 + B + 0$ .

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Step 3.5.2.1

Simplify the left side.

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Step 3.5.2.1.1

Remove parentheses.

$4 = 6 + B + 0$

$D = 0$

$C = 5$

$A = 3$

$F = 1$

$4 = 6 + B + 0$

$D = 0$

$C = 5$

$A = 3$

$F = 1$

Step 3.5.2.2

Simplify the right side.

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Step 3.5.2.2.1

Add $6 + B$ and $0$ .

$4 = 6 + B$

$D = 0$

$C = 5$

$A = 3$

$F = 1$

$4 = 6 + B$

$D = 0$

$C = 5$

$A = 3$

$F = 1$

$4 = 6 + B$

$D = 0$

$C = 5$

$A = 3$

$F = 1$

$4 = 6 + B$

$D = 0$

$C = 5$

$A = 3$

$F = 1$

Step 3.6

Solve for $B$ in $4 = 6 + B$ .

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Step 3.6.1

Rewrite the equation as $6 + B = 4$ .

$6 + B = 4$

$D = 0$

$C = 5$

$A = 3$

$F = 1$

Step 3.6.2

Move all terms not containing $B$ to the right side of the equation.

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Step 3.6.2.1

Subtract $6$ from both sides of the equation.

$B = 4 - 6$

$D = 0$

$C = 5$

$A = 3$

$F = 1$

Step 3.6.2.2

Subtract $6$ from $4$ .

$B = - 2$

$D = 0$

$C = 5$

$A = 3$

$F = 1$

$B = - 2$

$D = 0$

$C = 5$

$A = 3$

$F = 1$

$B = - 2$

$D = 0$

$C = 5$

$A = 3$

$F = 1$

Step 3.7

Solve the system of equations.

$B = - 2 D = 0 C = 5 A = 3 F = 1$

Step 3.8

List all of the solutions.

$B = - 2, D = 0, C = 5, A = 3, F = 1$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x} + \frac{B x + C}{{(x^{2} + 1)}^{2}} + \frac{D x + F}{x^{2} + 1}$ with the values found for $A$ , $B$ , $C$ , $D$ , and $F$ .

$\frac{3}{x} + \frac{- 2 x + 5}{{(x^{2} + 1)}^{2}} + \frac{0 x + 1}{x^{2} + 1}$

Step 5

Simplify.

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Step 5.1

Remove parentheses.

$\frac{3}{x} + \frac{(- 2) \cdot x + 5}{{(x^{2} + 1)}^{2}} + \frac{(0) \cdot x + 1}{x^{2} + 1}$

Step 5.2

Multiply $- 2$ by $x$ .

$\frac{3}{x} + \frac{- 2 x + 5}{{(x^{2} + 1)}^{2}} + \frac{(0) \cdot x + 1}{x^{2} + 1}$

Step 5.3

Multiply $0$ by $x$ .

$\frac{3}{x} + \frac{- 2 x + 5}{{(x^{2} + 1)}^{2}} + \frac{0 + 1}{x^{2} + 1}$

Step 5.4

Add $0$ and $1$ .

$\frac{3}{x} + \frac{- 2 x + 5}{{(x^{2} + 1)}^{2}} + \frac{1}{x^{2} + 1}$