Basic Math Examples

$\frac{t - 10}{t^{2} - t - 6} - \frac{t + 5}{t^{2} + 3 t + 2}$

Step 1

Factor $t^{2} + 3 t + 2$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Step 1.2

Write the factored form using these integers.

$\frac{t - 10}{t^{2} - t - 6} - \frac{t + 5}{(t + 1) (t + 2)}$

Step 2

To write $\frac{t - 10}{t^{2} - t - 6}$ as a fraction with a common denominator, multiply by $\frac{(t + 1) (t + 2)}{(t + 1) (t + 2)}$ .

$\frac{t - 10}{t^{2} - t - 6} \cdot \frac{(t + 1) (t + 2)}{(t + 1) (t + 2)} - \frac{t + 5}{(t + 1) (t + 2)}$

Step 3

To write $- \frac{t + 5}{(t + 1) (t + 2)}$ as a fraction with a common denominator, multiply by $\frac{t^{2} - t - 6}{t^{2} - t - 6}$ .

$\frac{t - 10}{t^{2} - t - 6} \cdot \frac{(t + 1) (t + 2)}{(t + 1) (t + 2)} - \frac{t + 5}{(t + 1) (t + 2)} \cdot \frac{t^{2} - t - 6}{t^{2} - t - 6}$

Step 4

Write each expression with a common denominator of $(t^{2} - t - 6) (t + 1) (t + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{t - 10}{t^{2} - t - 6}$ by $\frac{(t + 1) (t + 2)}{(t + 1) (t + 2)}$ .

$\frac{(t - 10) ((t + 1) (t + 2))}{(t^{2} - t - 6) ((t + 1) (t + 2))} - \frac{t + 5}{(t + 1) (t + 2)} \cdot \frac{t^{2} - t - 6}{t^{2} - t - 6}$

Step 4.2

Multiply $\frac{t + 5}{(t + 1) (t + 2)}$ by $\frac{t^{2} - t - 6}{t^{2} - t - 6}$ .

$\frac{(t - 10) ((t + 1) (t + 2))}{(t^{2} - t - 6) ((t + 1) (t + 2))} - \frac{(t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 4.3

Reorder the factors of $(t^{2} - t - 6) ((t + 1) (t + 2))$ .

$\frac{(t - 10) ((t + 1) (t + 2))}{(t + 1) (t + 2) (t^{2} - t - 6)} - \frac{(t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(t - 10) ((t + 1) (t + 2)) - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6

Rewrite $\frac{(t - 10) ((t + 1) (t + 2)) - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$ in a factored form.

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Step 6.1

Expand $(t + 1) (t + 2)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{(t - 10) (t (t + 2) + 1 (t + 2)) - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.1.2

Apply the distributive property.

$\frac{(t - 10) (t \cdot t + t \cdot 2 + 1 (t + 2)) - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.1.3

Apply the distributive property.

$\frac{(t - 10) (t \cdot t + t \cdot 2 + 1 t + 1 \cdot 2) - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $t$ by $t$ .

$\frac{(t - 10) (t^{2} + t \cdot 2 + 1 t + 1 \cdot 2) - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.2.1.2

Move $2$ to the left of $t$ .

$\frac{(t - 10) (t^{2} + 2 t + 1 t + 1 \cdot 2) - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.2.1.3

Multiply $t$ by $1$ .

$\frac{(t - 10) (t^{2} + 2 t + t + 1 \cdot 2) - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.2.1.4

Multiply $2$ by $1$ .

$\frac{(t - 10) (t^{2} + 2 t + t + 2) - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.2.2

Add $2 t$ and $t$ .

$\frac{(t - 10) (t^{2} + 3 t + 2) - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.3

Expand $(t - 10) (t^{2} + 3 t + 2)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{t \cdot t^{2} + t (3 t) + t \cdot 2 - 10 t^{2} - 10 (3 t) - 10 \cdot 2 - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.4

Simplify each term.

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Step 6.4.1

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

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Step 6.4.1.1

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

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Step 6.4.1.1.1

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

$\frac{t^{1} t^{2} + t (3 t) + t \cdot 2 - 10 t^{2} - 10 (3 t) - 10 \cdot 2 - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.4.1.1.2

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

$\frac{t^{1 + 2} + t (3 t) + t \cdot 2 - 10 t^{2} - 10 (3 t) - 10 \cdot 2 - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.4.1.2

Add $1$ and $2$ .

$\frac{t^{3} + t (3 t) + t \cdot 2 - 10 t^{2} - 10 (3 t) - 10 \cdot 2 - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.4.2

Rewrite using the commutative property of multiplication.

$\frac{t^{3} + 3 t \cdot t + t \cdot 2 - 10 t^{2} - 10 (3 t) - 10 \cdot 2 - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.4.3

Multiply $t$ by $t$ by adding the exponents.

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Step 6.4.3.1

Move $t$ .

$\frac{t^{3} + 3 (t \cdot t) + t \cdot 2 - 10 t^{2} - 10 (3 t) - 10 \cdot 2 - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.4.3.2

Multiply $t$ by $t$ .

$\frac{t^{3} + 3 t^{2} + t \cdot 2 - 10 t^{2} - 10 (3 t) - 10 \cdot 2 - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.4.4

Move $2$ to the left of $t$ .

$\frac{t^{3} + 3 t^{2} + 2 t - 10 t^{2} - 10 (3 t) - 10 \cdot 2 - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.4.5

Multiply $3$ by $- 10$ .

$\frac{t^{3} + 3 t^{2} + 2 t - 10 t^{2} - 30 t - 10 \cdot 2 - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.4.6

Multiply $- 10$ by $2$ .

$\frac{t^{3} + 3 t^{2} + 2 t - 10 t^{2} - 30 t - 20 - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.5

Subtract $10 t^{2}$ from $3 t^{2}$ .

$\frac{t^{3} - 7 t^{2} + 2 t - 30 t - 20 - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.6

Subtract $30 t$ from $2 t$ .

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - (t + 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.7

Apply the distributive property.

$\frac{t^{3} - 7 t^{2} - 28 t - 20 + (- t - 1 \cdot 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.8

Multiply $- 1$ by $5$ .

$\frac{t^{3} - 7 t^{2} - 28 t - 20 + (- t - 5) (t^{2} - t - 6)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.9

Expand $(- t - 5) (t^{2} - t - 6)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - t \cdot t^{2} - t (- t) - t \cdot - 6 - 5 t^{2} - 5 (- t) - 5 \cdot - 6}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.10

Simplify each term.

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Step 6.10.1

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

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Step 6.10.1.1

Move $t^{2}$ .

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - (t^{2} t) - t (- t) - t \cdot - 6 - 5 t^{2} - 5 (- t) - 5 \cdot - 6}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.10.1.2

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

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Step 6.10.1.2.1

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

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - (t^{2} t^{1}) - t (- t) - t \cdot - 6 - 5 t^{2} - 5 (- t) - 5 \cdot - 6}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.10.1.2.2

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

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - t^{2 + 1} - t (- t) - t \cdot - 6 - 5 t^{2} - 5 (- t) - 5 \cdot - 6}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.10.1.3

Add $2$ and $1$ .

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - t^{3} - t (- t) - t \cdot - 6 - 5 t^{2} - 5 (- t) - 5 \cdot - 6}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.10.2

Rewrite using the commutative property of multiplication.

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - t^{3} - 1 \cdot - 1 t \cdot t - t \cdot - 6 - 5 t^{2} - 5 (- t) - 5 \cdot - 6}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.10.3

Multiply $t$ by $t$ by adding the exponents.

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Step 6.10.3.1

Move $t$ .

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - t^{3} - 1 \cdot - 1 (t \cdot t) - t \cdot - 6 - 5 t^{2} - 5 (- t) - 5 \cdot - 6}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.10.3.2

Multiply $t$ by $t$ .

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - t^{3} - 1 \cdot - 1 t^{2} - t \cdot - 6 - 5 t^{2} - 5 (- t) - 5 \cdot - 6}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.10.4

Multiply $- 1$ by $- 1$ .

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - t^{3} + 1 t^{2} - t \cdot - 6 - 5 t^{2} - 5 (- t) - 5 \cdot - 6}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.10.5

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

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - t^{3} + t^{2} - t \cdot - 6 - 5 t^{2} - 5 (- t) - 5 \cdot - 6}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.10.6

Multiply $- 6$ by $- 1$ .

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - t^{3} + t^{2} + 6 t - 5 t^{2} - 5 (- t) - 5 \cdot - 6}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.10.7

Multiply $- 1$ by $- 5$ .

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - t^{3} + t^{2} + 6 t - 5 t^{2} + 5 t - 5 \cdot - 6}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.10.8

Multiply $- 5$ by $- 6$ .

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - t^{3} + t^{2} + 6 t - 5 t^{2} + 5 t + 30}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.11

Subtract $5 t^{2}$ from $t^{2}$ .

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - t^{3} - 4 t^{2} + 6 t + 5 t + 30}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.12

Add $6 t$ and $5 t$ .

$\frac{t^{3} - 7 t^{2} - 28 t - 20 - t^{3} - 4 t^{2} + 11 t + 30}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.13

Subtract $t^{3}$ from $t^{3}$ .

$\frac{- 7 t^{2} - 28 t - 20 + 0 - 4 t^{2} + 11 t + 30}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.14

Add $- 7 t^{2}$ and $0$ .

$\frac{- 7 t^{2} - 28 t - 20 - 4 t^{2} + 11 t + 30}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.15

Subtract $4 t^{2}$ from $- 7 t^{2}$ .

$\frac{- 11 t^{2} - 28 t - 20 + 11 t + 30}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.16

Add $- 28 t$ and $11 t$ .

$\frac{- 11 t^{2} - 17 t - 20 + 30}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.17

Add $- 20$ and $30$ .

$\frac{- 11 t^{2} - 17 t + 10}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.18

Rewrite $- 11 t^{2} - 17 t + 10$ in a factored form.

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Step 6.18.1

Factor $- 1$ out of $- 11 t^{2} - 17 t + 10$ .

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Step 6.18.1.1

Factor $- 1$ out of $- 11 t^{2}$ .

$\frac{- (11 t^{2}) - 17 t + 10}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.18.1.2

Factor $- 1$ out of $- 17 t$ .

$\frac{- (11 t^{2}) - (17 t) + 10}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.18.1.3

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{- (11 t^{2}) - (17 t) - 1 (- 10)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.18.1.4

Factor $- 1$ out of $- (11 t^{2}) - (17 t)$ .

$\frac{- (11 t^{2} + 17 t) - 1 (- 10)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.18.1.5

Factor $- 1$ out of $- (11 t^{2} + 17 t) - 1 (- 10)$ .

$\frac{- (11 t^{2} + 17 t - 10)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.18.2

Factor.

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Step 6.18.2.1

Factor by grouping.

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Step 6.18.2.1.1

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 $a \cdot c = 11 \cdot - 10 = - 110$ and whose sum is $b = 17$ .

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Step 6.18.2.1.1.1

Factor $17$ out of $17 t$ .

$\frac{- (11 t^{2} + 17 (t) - 10)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.18.2.1.1.2

Rewrite $17$ as $- 5$ plus $22$

$\frac{- (11 t^{2} + (- 5 + 22) t - 10)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.18.2.1.1.3

Apply the distributive property.

$\frac{- (11 t^{2} - 5 t + 22 t - 10)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.18.2.1.2

Factor out the greatest common factor from each group.

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Step 6.18.2.1.2.1

Group the first two terms and the last two terms.

$\frac{- ((11 t^{2} - 5 t) + 22 t - 10)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.18.2.1.2.2

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

$\frac{- (t (11 t - 5) + 2 (11 t - 5))}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.18.2.1.3

Factor the polynomial by factoring out the greatest common factor, $11 t - 5$ .

$\frac{- ((11 t - 5) (t + 2))}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.18.2.2

Remove unnecessary parentheses.

$\frac{- (11 t - 5) (t + 2)}{(t + 1) (t + 2) (t^{2} - t - 6)}$

Step 6.19

Factor.

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Step 6.19.1

Factor $t^{2} - t - 6$ using the AC method.

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Step 6.19.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 6.19.1.2

Write the factored form using these integers.

$\frac{- (11 t - 5) (t + 2)}{(t + 1) (t + 2) ((t - 3) (t + 2))}$

Step 6.19.2

Remove unnecessary parentheses.

$\frac{- (11 t - 5) (t + 2)}{(t + 1) (t + 2) (t - 3) (t + 2)}$

Step 6.20

Combine exponents.

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Step 6.20.1

Raise $t + 2$ to the power of $1$ .

$\frac{- (11 t - 5) (t + 2)}{(t + 1) ({(t + 2)}^{1} (t + 2)) (t - 3)}$

Step 6.20.2

Raise $t + 2$ to the power of $1$ .

$\frac{- (11 t - 5) (t + 2)}{(t + 1) ({(t + 2)}^{1} {(t + 2)}^{1}) (t - 3)}$

Step 6.20.3

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

$\frac{- (11 t - 5) (t + 2)}{(t + 1) {(t + 2)}^{1 + 1} (t - 3)}$

Step 6.20.4

Add $1$ and $1$ .

$\frac{- (11 t - 5) (t + 2)}{(t + 1) {(t + 2)}^{2} (t - 3)}$

Step 6.21

Reduce the expression $\frac{- (11 t - 5) (t + 2)}{(t + 1) {(t + 2)}^{2} (t - 3)}$ by cancelling the common factors.

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Step 6.21.1

Factor $t + 2$ out of $- (11 t - 5) (t + 2)$ .

$\frac{(t + 2) (- (11 t - 5))}{(t + 1) {(t + 2)}^{2} (t - 3)}$

Step 6.21.2

Factor $t + 2$ out of $(t + 1) {(t + 2)}^{2} (t - 3)$ .

$\frac{(t + 2) (- (11 t - 5))}{(t + 2) (((t + 1) (t + 2)) (t - 3))}$

Step 6.21.3

Cancel the common factor.

$\frac{(t + 2) (- (11 t - 5))}{(t + 2) (((t + 1) (t + 2)) (t - 3))}$

Step 6.21.4

Rewrite the expression.

$\frac{- (11 t - 5)}{((t + 1) (t + 2)) (t - 3)}$