Pre-Algebra Examples

$\frac{3 t - 22 t + 7}{3 t^{2} + 8 t - 3} \div \frac{2 t^{2} - 10 t + 1}{7 t^{2} + 20 t - 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 t - 22 t + 7}{3 t^{2} + 8 t - 3} \cdot \frac{7 t^{2} + 20 t - 3}{2 t^{2} - 10 t + 1}$

Step 2

Subtract $22 t$ from $3 t$ .

$\frac{- 19 t + 7}{3 t^{2} + 8 t - 3} \cdot \frac{7 t^{2} + 20 t - 3}{2 t^{2} - 10 t + 1}$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 3 \cdot - 3 = - 9$ and whose sum is $b = 8$ .

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Step 3.1.1

Factor $8$ out of $8 t$ .

$\frac{- 19 t + 7}{3 t^{2} + 8 (t) - 3} \cdot \frac{7 t^{2} + 20 t - 3}{2 t^{2} - 10 t + 1}$

Step 3.1.2

Rewrite $8$ as $- 1$ plus $9$

$\frac{- 19 t + 7}{3 t^{2} + (- 1 + 9) t - 3} \cdot \frac{7 t^{2} + 20 t - 3}{2 t^{2} - 10 t + 1}$

Step 3.1.3

Apply the distributive property.

$\frac{- 19 t + 7}{3 t^{2} - 1 t + 9 t - 3} \cdot \frac{7 t^{2} + 20 t - 3}{2 t^{2} - 10 t + 1}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{- 19 t + 7}{(3 t^{2} - 1 t) + 9 t - 3} \cdot \frac{7 t^{2} + 20 t - 3}{2 t^{2} - 10 t + 1}$

Step 3.2.2

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

$\frac{- 19 t + 7}{t (3 t - 1) + 3 (3 t - 1)} \cdot \frac{7 t^{2} + 20 t - 3}{2 t^{2} - 10 t + 1}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $3 t - 1$ .

$\frac{- 19 t + 7}{(3 t - 1) (t + 3)} \cdot \frac{7 t^{2} + 20 t - 3}{2 t^{2} - 10 t + 1}$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 7 \cdot - 3 = - 21$ and whose sum is $b = 20$ .

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Step 4.1.1

Factor $20$ out of $20 t$ .

$\frac{- 19 t + 7}{(3 t - 1) (t + 3)} \cdot \frac{7 t^{2} + 20 (t) - 3}{2 t^{2} - 10 t + 1}$

Step 4.1.2

Rewrite $20$ as $- 1$ plus $21$

$\frac{- 19 t + 7}{(3 t - 1) (t + 3)} \cdot \frac{7 t^{2} + (- 1 + 21) t - 3}{2 t^{2} - 10 t + 1}$

Step 4.1.3

Apply the distributive property.

$\frac{- 19 t + 7}{(3 t - 1) (t + 3)} \cdot \frac{7 t^{2} - 1 t + 21 t - 3}{2 t^{2} - 10 t + 1}$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$\frac{- 19 t + 7}{(3 t - 1) (t + 3)} \cdot \frac{(7 t^{2} - 1 t) + 21 t - 3}{2 t^{2} - 10 t + 1}$

Step 4.2.2

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

$\frac{- 19 t + 7}{(3 t - 1) (t + 3)} \cdot \frac{t (7 t - 1) + 3 (7 t - 1)}{2 t^{2} - 10 t + 1}$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $7 t - 1$ .

$\frac{- 19 t + 7}{(3 t - 1) (t + 3)} \cdot \frac{(7 t - 1) (t + 3)}{2 t^{2} - 10 t + 1}$

Step 5

Cancel the common factor of $t + 3$ .

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Step 5.1

Factor $t + 3$ out of $(3 t - 1) (t + 3)$ .

$\frac{- 19 t + 7}{(t + 3) (3 t - 1)} \cdot \frac{(7 t - 1) (t + 3)}{2 t^{2} - 10 t + 1}$

Step 5.2

Factor $t + 3$ out of $(7 t - 1) (t + 3)$ .

$\frac{- 19 t + 7}{(t + 3) (3 t - 1)} \cdot \frac{(t + 3) (7 t - 1)}{2 t^{2} - 10 t + 1}$

Step 5.3

Cancel the common factor.

$\frac{- 19 t + 7}{(t + 3) (3 t - 1)} \cdot \frac{(t + 3) (7 t - 1)}{2 t^{2} - 10 t + 1}$

Step 5.4

Rewrite the expression.

$\frac{- 19 t + 7}{3 t - 1} \cdot \frac{7 t - 1}{2 t^{2} - 10 t + 1}$

Step 6

Multiply $\frac{- 19 t + 7}{3 t - 1}$ by $\frac{7 t - 1}{2 t^{2} - 10 t + 1}$ .

$\frac{(- 19 t + 7) (7 t - 1)}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 7

Factor $- 1$ out of $- 19 t$ .

$\frac{(- (19 t) + 7) (7 t - 1)}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 8

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{(- (19 t) - 1 (- 7)) (7 t - 1)}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 9

Factor $- 1$ out of $- (19 t) - 1 (- 7)$ .

$\frac{- (19 t - 7) (7 t - 1)}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 10

Rewrite $- (19 t - 7)$ as $- 1 (19 t - 7)$ .

$\frac{- 1 (19 t - 7) (7 t - 1)}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 11

Move the negative in front of the fraction.

$- \frac{(19 t - 7) (7 t - 1)}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 12

Expand $(19 t - 7) (7 t - 1)$ using the FOIL Method.

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Step 12.1

Apply the distributive property.

$- \frac{19 t (7 t - 1) - 7 (7 t - 1)}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 12.2

Apply the distributive property.

$- \frac{19 t (7 t) + 19 t \cdot - 1 - 7 (7 t - 1)}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 12.3

Apply the distributive property.

$- \frac{19 t (7 t) + 19 t \cdot - 1 - 7 (7 t) - 7 \cdot - 1}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 13

Simplify and combine like terms.

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Step 13.1

Simplify each term.

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Step 13.1.1

Rewrite using the commutative property of multiplication.

$- \frac{19 \cdot 7 t \cdot t + 19 t \cdot - 1 - 7 (7 t) - 7 \cdot - 1}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 13.1.2

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

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Step 13.1.2.1

Move $t$ .

$- \frac{19 \cdot 7 (t \cdot t) + 19 t \cdot - 1 - 7 (7 t) - 7 \cdot - 1}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 13.1.2.2

Multiply $t$ by $t$ .

$- \frac{19 \cdot 7 t^{2} + 19 t \cdot - 1 - 7 (7 t) - 7 \cdot - 1}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 13.1.3

Multiply $19$ by $7$ .

$- \frac{133 t^{2} + 19 t \cdot - 1 - 7 (7 t) - 7 \cdot - 1}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 13.1.4

Multiply $- 1$ by $19$ .

$- \frac{133 t^{2} - 19 t - 7 (7 t) - 7 \cdot - 1}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 13.1.5

Multiply $7$ by $- 7$ .

$- \frac{133 t^{2} - 19 t - 49 t - 7 \cdot - 1}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 13.1.6

Multiply $- 7$ by $- 1$ .

$- \frac{133 t^{2} - 19 t - 49 t + 7}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 13.2

Subtract $49 t$ from $- 19 t$ .

$- \frac{133 t^{2} - 68 t + 7}{(3 t - 1) (2 t^{2} - 10 t + 1)}$

Step 14

Split the fraction $\frac{133 t^{2} - 68 t + 7}{(3 t - 1) (2 t^{2} - 10 t + 1)}$ into two fractions.

$- (\frac{133 t^{2} - 68 t}{(3 t - 1) (2 t^{2} - 10 t + 1)} + \frac{7}{(3 t - 1) (2 t^{2} - 10 t + 1)})$

Step 15

Split the fraction $\frac{133 t^{2} - 68 t}{(3 t - 1) (2 t^{2} - 10 t + 1)}$ into two fractions.

$- (\frac{133 t^{2}}{(3 t - 1) (2 t^{2} - 10 t + 1)} + \frac{- 68 t}{(3 t - 1) (2 t^{2} - 10 t + 1)} + \frac{7}{(3 t - 1) (2 t^{2} - 10 t + 1)})$

Step 16

Move the negative in front of the fraction.

$- (\frac{133 t^{2}}{(3 t - 1) (2 t^{2} - 10 t + 1)} - \frac{68 t}{(3 t - 1) (2 t^{2} - 10 t + 1)} + \frac{7}{(3 t - 1) (2 t^{2} - 10 t + 1)})$