Basic Math Examples

$\frac{3 t^{3} - 14 t^{2} + 8 t}{20 + 3 t - 2 t^{2}} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $t$ out of $3 t^{3} - 14 t^{2} + 8 t$ .

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Step 1.1.1

Factor $t$ out of $3 t^{3}$ .

$\frac{t (3 t^{2}) - 14 t^{2} + 8 t}{20 + 3 t - 2 t^{2}} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 1.1.2

Factor $t$ out of $- 14 t^{2}$ .

$\frac{t (3 t^{2}) + t (- 14 t) + 8 t}{20 + 3 t - 2 t^{2}} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 1.1.3

Factor $t$ out of $8 t$ .

$\frac{t (3 t^{2}) + t (- 14 t) + t \cdot 8}{20 + 3 t - 2 t^{2}} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 1.1.4

Factor $t$ out of $t (3 t^{2}) + t (- 14 t)$ .

$\frac{t (3 t^{2} - 14 t) + t \cdot 8}{20 + 3 t - 2 t^{2}} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 1.1.5

Factor $t$ out of $t (3 t^{2} - 14 t) + t \cdot 8$ .

$\frac{t (3 t^{2} - 14 t + 8)}{20 + 3 t - 2 t^{2}} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 1.2

Factor by grouping.

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Step 1.2.1

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 $a \cdot c = 3 \cdot 8 = 24$ and whose sum is $b = - 14$ .

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Step 1.2.1.1

Factor $- 14$ out of $- 14 t$ .

$\frac{t (3 t^{2} - 14 (t) + 8)}{20 + 3 t - 2 t^{2}} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 1.2.1.2

Rewrite $- 14$ as $- 2$ plus $- 12$

$\frac{t (3 t^{2} + (- 2 - 12) t + 8)}{20 + 3 t - 2 t^{2}} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 1.2.1.3

Apply the distributive property.

$\frac{t (3 t^{2} - 2 t - 12 t + 8)}{20 + 3 t - 2 t^{2}} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{t ((3 t^{2} - 2 t) - 12 t + 8)}{20 + 3 t - 2 t^{2}} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 1.2.2.2

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

$\frac{t (t (3 t - 2) - 4 (3 t - 2))}{20 + 3 t - 2 t^{2}} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $3 t - 2$ .

$\frac{t ((3 t - 2) (t - 4))}{20 + 3 t - 2 t^{2}} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

$\frac{t (3 t - 2) (t - 4)}{20 + 3 t - 2 t^{2}} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 2

Factor by grouping.

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Step 2.1

Reorder terms.

$\frac{t (3 t - 2) (t - 4)}{- 2 t^{2} + 3 t + 20} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 2.2

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 $a \cdot c = - 2 \cdot 20 = - 40$ and whose sum is $b = 3$ .

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Step 2.2.1

Factor $3$ out of $3 t$ .

$\frac{t (3 t - 2) (t - 4)}{- 2 t^{2} + 3 (t) + 20} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 2.2.2

Rewrite $3$ as $- 5$ plus $8$

$\frac{t (3 t - 2) (t - 4)}{- 2 t^{2} + (- 5 + 8) t + 20} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 2.2.3

Apply the distributive property.

$\frac{t (3 t - 2) (t - 4)}{- 2 t^{2} - 5 t + 8 t + 20} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 2.3

Factor out the greatest common factor from each group.

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Step 2.3.1

Group the first two terms and the last two terms.

$\frac{t (3 t - 2) (t - 4)}{(- 2 t^{2} - 5 t) + 8 t + 20} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 2.3.2

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

$\frac{t (3 t - 2) (t - 4)}{t (- 2 t - 5) - 4 (- 2 t - 5)} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

Step 2.4

Factor the polynomial by factoring out the greatest common factor, $- 2 t - 5$ .

$\frac{t (3 t - 2) (t - 4)}{(- 2 t - 5) (t - 4)} \cdot \frac{16 t^{2} + 34 t - 15}{24 t^{2} - 25 t + 6}$

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 $a \cdot c = 16 \cdot - 15 = - 240$ and whose sum is $b = 34$ .

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Step 3.1.1

Factor $34$ out of $34 t$ .

$\frac{t (3 t - 2) (t - 4)}{(- 2 t - 5) (t - 4)} \cdot \frac{16 t^{2} + 34 (t) - 15}{24 t^{2} - 25 t + 6}$

Step 3.1.2

Rewrite $34$ as $- 6$ plus $40$

$\frac{t (3 t - 2) (t - 4)}{(- 2 t - 5) (t - 4)} \cdot \frac{16 t^{2} + (- 6 + 40) t - 15}{24 t^{2} - 25 t + 6}$

Step 3.1.3

Apply the distributive property.

$\frac{t (3 t - 2) (t - 4)}{(- 2 t - 5) (t - 4)} \cdot \frac{16 t^{2} - 6 t + 40 t - 15}{24 t^{2} - 25 t + 6}$

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.

$\frac{t (3 t - 2) (t - 4)}{(- 2 t - 5) (t - 4)} \cdot \frac{(16 t^{2} - 6 t) + 40 t - 15}{24 t^{2} - 25 t + 6}$

Step 3.2.2

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

$\frac{t (3 t - 2) (t - 4)}{(- 2 t - 5) (t - 4)} \cdot \frac{2 t (8 t - 3) + 5 (8 t - 3)}{24 t^{2} - 25 t + 6}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $8 t - 3$ .

$\frac{t (3 t - 2) (t - 4)}{(- 2 t - 5) (t - 4)} \cdot \frac{(8 t - 3) (2 t + 5)}{24 t^{2} - 25 t + 6}$

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 $a \cdot c = 24 \cdot 6 = 144$ and whose sum is $b = - 25$ .

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Step 4.1.1

Factor $- 25$ out of $- 25 t$ .

$\frac{t (3 t - 2) (t - 4)}{(- 2 t - 5) (t - 4)} \cdot \frac{(8 t - 3) (2 t + 5)}{24 t^{2} - 25 (t) + 6}$

Step 4.1.2

Rewrite $- 25$ as $- 9$ plus $- 16$

$\frac{t (3 t - 2) (t - 4)}{(- 2 t - 5) (t - 4)} \cdot \frac{(8 t - 3) (2 t + 5)}{24 t^{2} + (- 9 - 16) t + 6}$

Step 4.1.3

Apply the distributive property.

$\frac{t (3 t - 2) (t - 4)}{(- 2 t - 5) (t - 4)} \cdot \frac{(8 t - 3) (2 t + 5)}{24 t^{2} - 9 t - 16 t + 6}$

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.

$\frac{t (3 t - 2) (t - 4)}{(- 2 t - 5) (t - 4)} \cdot \frac{(8 t - 3) (2 t + 5)}{(24 t^{2} - 9 t) - 16 t + 6}$

Step 4.2.2

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

$\frac{t (3 t - 2) (t - 4)}{(- 2 t - 5) (t - 4)} \cdot \frac{(8 t - 3) (2 t + 5)}{3 t (8 t - 3) - 2 (8 t - 3)}$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $8 t - 3$ .

$\frac{t (3 t - 2) (t - 4)}{(- 2 t - 5) (t - 4)} \cdot \frac{(8 t - 3) (2 t + 5)}{(8 t - 3) (3 t - 2)}$

Step 5

Cancel the common factor of $3 t - 2$ .

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Step 5.1

Factor $3 t - 2$ out of $t (3 t - 2) (t - 4)$ .

$\frac{(3 t - 2) (t (t - 4))}{(- 2 t - 5) (t - 4)} \cdot \frac{(8 t - 3) (2 t + 5)}{(8 t - 3) (3 t - 2)}$

Step 5.2

Factor $3 t - 2$ out of $(8 t - 3) (3 t - 2)$ .

$\frac{(3 t - 2) (t (t - 4))}{(- 2 t - 5) (t - 4)} \cdot \frac{(8 t - 3) (2 t + 5)}{(3 t - 2) (8 t - 3)}$

Step 5.3

Cancel the common factor.

$\frac{(3 t - 2) (t (t - 4))}{(- 2 t - 5) (t - 4)} \cdot \frac{(8 t - 3) (2 t + 5)}{(3 t - 2) (8 t - 3)}$

Step 5.4

Rewrite the expression.

$\frac{t (t - 4)}{(- 2 t - 5) (t - 4)} \cdot \frac{(8 t - 3) (2 t + 5)}{8 t - 3}$

Step 6

Multiply $\frac{t (t - 4)}{(- 2 t - 5) (t - 4)}$ by $\frac{(8 t - 3) (2 t + 5)}{8 t - 3}$ .

$\frac{t (t - 4) ((8 t - 3) (2 t + 5))}{(- 2 t - 5) (t - 4) (8 t - 3)}$

Step 7

Cancel the common factor of $t - 4$ .

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Step 7.1

Cancel the common factor.

$\frac{t (t - 4) ((8 t - 3) (2 t + 5))}{(- 2 t - 5) (t - 4) (8 t - 3)}$

Step 7.2

Rewrite the expression.

$\frac{t ((8 t - 3) (2 t + 5))}{(- 2 t - 5) (8 t - 3)}$

Step 8

Cancel the common factor of $8 t - 3$ .

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Step 8.1

Cancel the common factor.

$\frac{t ((8 t - 3) (2 t + 5))}{(- 2 t - 5) (8 t - 3)}$

Step 8.2

Rewrite the expression.

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

Step 9

Cancel the common factor of $2 t + 5$ and $- 2 t - 5$ .

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Step 9.1

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

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

Step 9.2

Rewrite $5$ as $- 1 (- 5)$ .

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

Step 9.3

Factor $- 1$ out of $- (- 2 t) - 1 (- 5)$ .

$\frac{t (- (- 2 t - 5))}{- 2 t - 5}$

Step 9.4

Rewrite $- (- 2 t - 5)$ as $- 1 (- 2 t - 5)$ .

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

Step 9.5

Cancel the common factor.

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

Step 9.6

Divide $t \cdot (- 1)$ by $1$ .

$t \cdot (- 1)$

Step 10

Simplify the expression.

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Step 10.1

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

$- 1 t$

Step 10.2

Rewrite $- 1 t$ as $- t$ .

$- t$