Algebra Examples

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

Step 1

Decompose the fraction and multiply through by the common denominator.

Tap for more steps...

Step 1.1

Factor the fraction.

Tap for more steps...

Step 1.1.1

Factor by grouping.

Tap for more steps...

Step 1.1.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = 1$ .

Tap for more steps...

Step 1.1.1.1.1

Multiply by $1$ .

$\frac{3 x^{2} + 1 x - 4}{x^{2} - 2 x}$

Step 1.1.1.1.2

Rewrite $1$ as $- 3$ plus $4$

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

Step 1.1.1.1.3

Apply the distributive property.

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

Step 1.1.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.1.1.2.1

Group the first two terms and the last two terms.

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

Step 1.1.1.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 1.1.1.3

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

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

Step 1.1.2

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

Tap for more steps...

Step 1.1.2.1

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

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

Step 1.1.2.2

Factor $x$ out of $- 2 x$ .

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

Step 1.1.2.3

Factor $x$ out of $x \cdot x + x \cdot - 2$ .

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

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 in the denominator is linear, put a single variable in its place $B$ .

$\frac{A}{x} + \frac{B}{x - 2}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $x (x - 2)$ .

$\frac{(x - 1) (3 x + 4) (x (x - 2))}{x (x - 2)} = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.4

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 1.4.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.4.1.1

Cancel the common factor.

$\frac{(x - 1) (3 x + 4) (x (x - 2))}{x (x - 2)} = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.4.1.2

Rewrite the expression.

$\frac{(x - 1) (3 x + 4) (x - 2)}{x - 2} = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.4.2

Cancel the common factor of $x - 2$ .

Tap for more steps...

Step 1.4.2.1

Cancel the common factor.

$\frac{(x - 1) (3 x + 4) (x - 2)}{x - 2} = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.4.2.2

Divide $(x - 1) (3 x + 4)$ by $1$ .

$(x - 1) (3 x + 4) = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.5

Expand $(x - 1) (3 x + 4)$ using the FOIL Method.

Tap for more steps...

Step 1.5.1

Apply the distributive property.

$x (3 x + 4) - 1 (3 x + 4) = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.5.2

Apply the distributive property.

$x (3 x) + x \cdot 4 - 1 (3 x + 4) = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.5.3

Apply the distributive property.

$x (3 x) + x \cdot 4 - 1 (3 x) - 1 \cdot 4 = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.6

Simplify and combine like terms.

Tap for more steps...

Step 1.6.1

Simplify each term.

Tap for more steps...

Step 1.6.1.1

Rewrite using the commutative property of multiplication.

$3 x \cdot x + x \cdot 4 - 1 (3 x) - 1 \cdot 4 = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.6.1.2

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

Tap for more steps...

Step 1.6.1.2.1

Move $x$ .

$3 (x \cdot x) + x \cdot 4 - 1 (3 x) - 1 \cdot 4 = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.6.1.2.2

Multiply $x$ by $x$ .

$3 x^{2} + x \cdot 4 - 1 (3 x) - 1 \cdot 4 = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.6.1.3

Move $4$ to the left of $x$ .

$3 x^{2} + 4 \cdot x - 1 (3 x) - 1 \cdot 4 = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.6.1.4

Multiply $3$ by $- 1$ .

$3 x^{2} + 4 x - 3 x - 1 \cdot 4 = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.6.1.5

Multiply $- 1$ by $4$ .

$3 x^{2} + 4 x - 3 x - 4 = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.6.2

Subtract $3 x$ from $4 x$ .

$3 x^{2} + x - 4 = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.7

Simplify each term.

Tap for more steps...

Step 1.7.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.7.1.1

Cancel the common factor.

$3 x^{2} + x - 4 = \frac{A (x (x - 2))}{x} + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.7.1.2

Divide $A (x - 2)$ by $1$ .

$3 x^{2} + x - 4 = A (x - 2) + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.7.2

Apply the distributive property.

$3 x^{2} + x - 4 = A x + A \cdot - 2 + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.7.3

Move $- 2$ to the left of $A$ .

$3 x^{2} + x - 4 = A x - 2 \cdot A + \frac{(B) (x (x - 2))}{x - 2}$

Step 1.7.4

Cancel the common factor of $x - 2$ .

Tap for more steps...

Step 1.7.4.1

Cancel the common factor.

$3 x^{2} + x - 4 = A x - 2 A + \frac{B (x (x - 2))}{x - 2}$

Step 1.7.4.2

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

$3 x^{2} + x - 4 = A x - 2 A + (B) (x)$

$3 x^{2} + x - 4 = A x - 2 A + B x$

Step 1.8

Move $- 2 A$ .

$3 x^{2} + x - 4 = A x + B x - 2 A$

Step 2

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

Tap for more steps...

Step 2.1

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.

$1 = A + B$

Step 2.2

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.

$- 4 = - 2 A$

Step 2.3

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

$1 = A + B$

$- 4 = - 2 A$

$1 = A + B$

$- 4 = - 2 A$

Step 3

Solve the system of equations.

Tap for more steps...

Step 3.1

Solve for $A$ in $- 4 = - 2 A$ .

Tap for more steps...

Step 3.1.1

Rewrite the equation as $- 2 A = - 4$ .

$- 2 A = - 4$

$1 = A + B$

Step 3.1.2

Divide each term in $- 2 A = - 4$ by $- 2$ and simplify.

Tap for more steps...

Step 3.1.2.1

Divide each term in $- 2 A = - 4$ by $- 2$ .

$\frac{- 2 A}{- 2} = \frac{- 4}{- 2}$

$1 = A + B$

Step 3.1.2.2

Simplify the left side.

Tap for more steps...

Step 3.1.2.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{- 2 A}{- 2} = \frac{- 4}{- 2}$

$1 = A + B$

Step 3.1.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{- 4}{- 2}$

$1 = A + B$

$A = \frac{- 4}{- 2}$

$1 = A + B$

$A = \frac{- 4}{- 2}$

$1 = A + B$

Step 3.1.2.3

Simplify the right side.

Tap for more steps...

Step 3.1.2.3.1

Divide $- 4$ by $- 2$ .

$A = 2$

$1 = A + B$

$A = 2$

$1 = A + B$

$A = 2$

$1 = A + B$

$A = 2$

$1 = A + B$

Step 3.2

Replace all occurrences of $A$ with $2$ in each equation.

Tap for more steps...

Step 3.2.1

Replace all occurrences of $A$ in $1 = A + B$ with $2$ .

$1 = (2) + B$

$A = 2$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Remove parentheses.

$1 = 2 + B$

$A = 2$

$1 = 2 + B$

$A = 2$

$1 = 2 + B$

$A = 2$

Step 3.3

Solve for $B$ in $1 = 2 + B$ .

Tap for more steps...

Step 3.3.1

Rewrite the equation as $2 + B = 1$ .

$2 + B = 1$

$A = 2$

Step 3.3.2

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

Tap for more steps...

Step 3.3.2.1

Subtract $2$ from both sides of the equation.

$B = 1 - 2$

$A = 2$

Step 3.3.2.2

Subtract $2$ from $1$ .

$B = - 1$

$A = 2$

$B = - 1$

$A = 2$

$B = - 1$

$A = 2$

Step 3.4

Solve the system of equations.

$B = - 1 A = 2$

Step 3.5

List all of the solutions.

$B = - 1, A = 2$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x} + \frac{B}{x - 2}$ with the values found for $A$ and $B$ .

$\frac{2}{x} + \frac{- 1}{x - 2}$