Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor by grouping.

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Step 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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 1$ .

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Step 1.1.1.1

Factor $- 1$ out of $- k$ .

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

Step 1.1.1.2

Rewrite $- 1$ as $2$ plus $- 3$

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

Step 1.1.1.3

Apply the distributive property.

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

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

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

Step 1.1.2.2

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

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

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $k + 1$ .

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

Step 1.2

Cancel the common factor of $k + 1$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 2

To write $\frac{k^{2}}{(k + 1) (k - 1)}$ as a fraction with a common denominator, multiply by $\frac{k + 2}{k + 2}$ .

$\frac{k^{2}}{(k + 1) (k - 1)} \cdot \frac{k + 2}{k + 2} - \frac{2 k - 3}{k + 2}$

Step 3

To write $- \frac{2 k - 3}{k + 2}$ as a fraction with a common denominator, multiply by $\frac{(k + 1) (k - 1)}{(k + 1) (k - 1)}$ .

$\frac{k^{2}}{(k + 1) (k - 1)} \cdot \frac{k + 2}{k + 2} - \frac{2 k - 3}{k + 2} \cdot \frac{(k + 1) (k - 1)}{(k + 1) (k - 1)}$

Step 4

Write each expression with a common denominator of $(k + 1) (k - 1) (k + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{k^{2}}{(k + 1) (k - 1)}$ by $\frac{k + 2}{k + 2}$ .

$\frac{k^{2} (k + 2)}{(k + 1) (k - 1) (k + 2)} - \frac{2 k - 3}{k + 2} \cdot \frac{(k + 1) (k - 1)}{(k + 1) (k - 1)}$

Step 4.2

Multiply $\frac{2 k - 3}{k + 2}$ by $\frac{(k + 1) (k - 1)}{(k + 1) (k - 1)}$ .

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

Step 4.3

Reorder the factors of $(k + 1) (k - 1) (k + 2)$ .

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

Step 4.4

Reorder the factors of $(k + 2) ((k + 1) (k - 1))$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{k^{2} k + k^{2} \cdot 2 - (2 k - 3) ((k + 1) (k - 1))}{(k + 1) (k + 2) (k - 1)}$

Step 6.2

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

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Step 6.2.1

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

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Step 6.2.1.1

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

$\frac{k^{2} k^{1} + k^{2} \cdot 2 - (2 k - 3) ((k + 1) (k - 1))}{(k + 1) (k + 2) (k - 1)}$

Step 6.2.1.2

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

$\frac{k^{2 + 1} + k^{2} \cdot 2 - (2 k - 3) ((k + 1) (k - 1))}{(k + 1) (k + 2) (k - 1)}$

Step 6.2.2

Add $2$ and $1$ .

$\frac{k^{3} + k^{2} \cdot 2 - (2 k - 3) ((k + 1) (k - 1))}{(k + 1) (k + 2) (k - 1)}$

Step 6.3

Move $2$ to the left of $k^{2}$ .

$\frac{k^{3} + 2 \cdot k^{2} - (2 k - 3) ((k + 1) (k - 1))}{(k + 1) (k + 2) (k - 1)}$

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $2$ by $- 1$ .

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

Step 6.6

Multiply $- 1$ by $- 3$ .

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

Step 6.7

Expand $(k + 1) (k - 1)$ using the FOIL Method.

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Step 6.7.1

Apply the distributive property.

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

Step 6.7.2

Apply the distributive property.

$\frac{k^{3} + 2 k^{2} + (- 2 k + 3) (k \cdot k + k \cdot - 1 + 1 (k - 1))}{(k + 1) (k + 2) (k - 1)}$

Step 6.7.3

Apply the distributive property.

$\frac{k^{3} + 2 k^{2} + (- 2 k + 3) (k \cdot k + k \cdot - 1 + 1 k + 1 \cdot - 1)}{(k + 1) (k + 2) (k - 1)}$

Step 6.8

Simplify and combine like terms.

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Step 6.8.1

Simplify each term.

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Step 6.8.1.1

Multiply $k$ by $k$ .

$\frac{k^{3} + 2 k^{2} + (- 2 k + 3) (k^{2} + k \cdot - 1 + 1 k + 1 \cdot - 1)}{(k + 1) (k + 2) (k - 1)}$

Step 6.8.1.2

Move $- 1$ to the left of $k$ .

$\frac{k^{3} + 2 k^{2} + (- 2 k + 3) (k^{2} - 1 \cdot k + 1 k + 1 \cdot - 1)}{(k + 1) (k + 2) (k - 1)}$

Step 6.8.1.3

Rewrite $- 1 k$ as $- k$ .

$\frac{k^{3} + 2 k^{2} + (- 2 k + 3) (k^{2} - k + 1 k + 1 \cdot - 1)}{(k + 1) (k + 2) (k - 1)}$

Step 6.8.1.4

Multiply $k$ by $1$ .

$\frac{k^{3} + 2 k^{2} + (- 2 k + 3) (k^{2} - k + k + 1 \cdot - 1)}{(k + 1) (k + 2) (k - 1)}$

Step 6.8.1.5

Multiply $- 1$ by $1$ .

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

Step 6.8.2

Add $- k$ and $k$ .

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

Step 6.8.3

Add $k^{2}$ and $0$ .

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

Step 6.9

Expand $(- 2 k + 3) (k^{2} - 1)$ using the FOIL Method.

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Step 6.9.1

Apply the distributive property.

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

Step 6.9.2

Apply the distributive property.

$\frac{k^{3} + 2 k^{2} - 2 k \cdot k^{2} - 2 k \cdot - 1 + 3 (k^{2} - 1)}{(k + 1) (k + 2) (k - 1)}$

Step 6.9.3

Apply the distributive property.

$\frac{k^{3} + 2 k^{2} - 2 k \cdot k^{2} - 2 k \cdot - 1 + 3 k^{2} + 3 \cdot - 1}{(k + 1) (k + 2) (k - 1)}$

Step 6.10

Simplify each term.

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Step 6.10.1

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

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Step 6.10.1.1

Move $k^{2}$ .

$\frac{k^{3} + 2 k^{2} - 2 (k^{2} k) - 2 k \cdot - 1 + 3 k^{2} + 3 \cdot - 1}{(k + 1) (k + 2) (k - 1)}$

Step 6.10.1.2

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

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Step 6.10.1.2.1

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

$\frac{k^{3} + 2 k^{2} - 2 (k^{2} k^{1}) - 2 k \cdot - 1 + 3 k^{2} + 3 \cdot - 1}{(k + 1) (k + 2) (k - 1)}$

Step 6.10.1.2.2

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

$\frac{k^{3} + 2 k^{2} - 2 k^{2 + 1} - 2 k \cdot - 1 + 3 k^{2} + 3 \cdot - 1}{(k + 1) (k + 2) (k - 1)}$

Step 6.10.1.3

Add $2$ and $1$ .

$\frac{k^{3} + 2 k^{2} - 2 k^{3} - 2 k \cdot - 1 + 3 k^{2} + 3 \cdot - 1}{(k + 1) (k + 2) (k - 1)}$

Step 6.10.2

Multiply $- 1$ by $- 2$ .

$\frac{k^{3} + 2 k^{2} - 2 k^{3} + 2 k + 3 k^{2} + 3 \cdot - 1}{(k + 1) (k + 2) (k - 1)}$

Step 6.10.3

Multiply $3$ by $- 1$ .

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

Step 6.11

Subtract $2 k^{3}$ from $k^{3}$ .

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

Step 6.12

Add $2 k^{2}$ and $3 k^{2}$ .

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

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- k^{3}$ .

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

Step 7.2

Factor $- 1$ out of $5 k^{2}$ .

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

Step 7.3

Factor $- 1$ out of $- (k^{3}) - (- 5 k^{2})$ .

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

Step 7.4

Factor $- 1$ out of $2 k$ .

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

Step 7.5

Factor $- 1$ out of $- (k^{3} - 5 k^{2}) - (- 2 k)$ .

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

Step 7.6

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 7.7

Factor $- 1$ out of $- (k^{3} - 5 k^{2} - 2 k) - 1 (3)$ .

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

Step 7.8

Simplify the expression.

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Step 7.8.1

Rewrite $- (k^{3} - 5 k^{2} - 2 k + 3)$ as $- 1 (k^{3} - 5 k^{2} - 2 k + 3)$ .

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

Step 7.8.2

Move the negative in front of the fraction.

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