Basic Math Examples

$\frac{8 t^{2} - 43 t + 15}{8 t^{2} + 61 t - 24} \cdot \frac{5 t^{2} + 37 t - 24}{40 t - 39 t + 9}$

Step 1

Factor by grouping.

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Step 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 $a \cdot c = 8 \cdot 15 = 120$ and whose sum is $b = - 43$ .

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Step 1.1.1

Factor $- 43$ out of $- 43 t$ .

$\frac{8 t^{2} - 43 (t) + 15}{8 t^{2} + 61 t - 24} \cdot \frac{5 t^{2} + 37 t - 24}{40 t - 39 t + 9}$

Step 1.1.2

Rewrite $- 43$ as $- 3$ plus $- 40$

$\frac{8 t^{2} + (- 3 - 40) t + 15}{8 t^{2} + 61 t - 24} \cdot \frac{5 t^{2} + 37 t - 24}{40 t - 39 t + 9}$

Step 1.1.3

Apply the distributive property.

$\frac{8 t^{2} - 3 t - 40 t + 15}{8 t^{2} + 61 t - 24} \cdot \frac{5 t^{2} + 37 t - 24}{40 t - 39 t + 9}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(8 t^{2} - 3 t) - 40 t + 15}{8 t^{2} + 61 t - 24} \cdot \frac{5 t^{2} + 37 t - 24}{40 t - 39 t + 9}$

Step 1.2.2

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

$\frac{t (8 t - 3) - 5 (8 t - 3)}{8 t^{2} + 61 t - 24} \cdot \frac{5 t^{2} + 37 t - 24}{40 t - 39 t + 9}$

Step 1.3

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

$\frac{(8 t - 3) (t - 5)}{8 t^{2} + 61 t - 24} \cdot \frac{5 t^{2} + 37 t - 24}{40 t - 39 t + 9}$

Step 2

Factor by grouping.

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Step 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 $a \cdot c = 8 \cdot - 24 = - 192$ and whose sum is $b = 61$ .

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Step 2.1.1

Factor $61$ out of $61 t$ .

$\frac{(8 t - 3) (t - 5)}{8 t^{2} + 61 (t) - 24} \cdot \frac{5 t^{2} + 37 t - 24}{40 t - 39 t + 9}$

Step 2.1.2

Rewrite $61$ as $- 3$ plus $64$

$\frac{(8 t - 3) (t - 5)}{8 t^{2} + (- 3 + 64) t - 24} \cdot \frac{5 t^{2} + 37 t - 24}{40 t - 39 t + 9}$

Step 2.1.3

Apply the distributive property.

$\frac{(8 t - 3) (t - 5)}{8 t^{2} - 3 t + 64 t - 24} \cdot \frac{5 t^{2} + 37 t - 24}{40 t - 39 t + 9}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(8 t - 3) (t - 5)}{(8 t^{2} - 3 t) + 64 t - 24} \cdot \frac{5 t^{2} + 37 t - 24}{40 t - 39 t + 9}$

Step 2.2.2

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

$\frac{(8 t - 3) (t - 5)}{t (8 t - 3) + 8 (8 t - 3)} \cdot \frac{5 t^{2} + 37 t - 24}{40 t - 39 t + 9}$

Step 2.3

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

$\frac{(8 t - 3) (t - 5)}{(8 t - 3) (t + 8)} \cdot \frac{5 t^{2} + 37 t - 24}{40 t - 39 t + 9}$

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 $a \cdot c = 5 \cdot - 24 = - 120$ and whose sum is $b = 37$ .

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Step 3.1.1

Factor $37$ out of $37 t$ .

$\frac{(8 t - 3) (t - 5)}{(8 t - 3) (t + 8)} \cdot \frac{5 t^{2} + 37 (t) - 24}{40 t - 39 t + 9}$

Step 3.1.2

Rewrite $37$ as $- 3$ plus $40$

$\frac{(8 t - 3) (t - 5)}{(8 t - 3) (t + 8)} \cdot \frac{5 t^{2} + (- 3 + 40) t - 24}{40 t - 39 t + 9}$

Step 3.1.3

Apply the distributive property.

$\frac{(8 t - 3) (t - 5)}{(8 t - 3) (t + 8)} \cdot \frac{5 t^{2} - 3 t + 40 t - 24}{40 t - 39 t + 9}$

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.

$\frac{(8 t - 3) (t - 5)}{(8 t - 3) (t + 8)} \cdot \frac{(5 t^{2} - 3 t) + 40 t - 24}{40 t - 39 t + 9}$

Step 3.2.2

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

$\frac{(8 t - 3) (t - 5)}{(8 t - 3) (t + 8)} \cdot \frac{t (5 t - 3) + 8 (5 t - 3)}{40 t - 39 t + 9}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $5 t - 3$ .

$\frac{(8 t - 3) (t - 5)}{(8 t - 3) (t + 8)} \cdot \frac{(5 t - 3) (t + 8)}{40 t - 39 t + 9}$

Step 4

Subtract $39 t$ from $40 t$ .

$\frac{(8 t - 3) (t - 5)}{(8 t - 3) (t + 8)} \cdot \frac{(5 t - 3) (t + 8)}{t + 9}$

Step 5

Cancel the common factor of $t + 8$ .

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Step 5.1

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

$\frac{(8 t - 3) (t - 5)}{(t + 8) (8 t - 3)} \cdot \frac{(5 t - 3) (t + 8)}{t + 9}$

Step 5.2

Factor $t + 8$ out of $(5 t - 3) (t + 8)$ .

$\frac{(8 t - 3) (t - 5)}{(t + 8) (8 t - 3)} \cdot \frac{(t + 8) (5 t - 3)}{t + 9}$

Step 5.3

Cancel the common factor.

$\frac{(8 t - 3) (t - 5)}{(t + 8) (8 t - 3)} \cdot \frac{(t + 8) (5 t - 3)}{t + 9}$

Step 5.4

Rewrite the expression.

$\frac{(8 t - 3) (t - 5)}{8 t - 3} \cdot \frac{5 t - 3}{t + 9}$

Step 6

Multiply $\frac{(8 t - 3) (t - 5)}{8 t - 3}$ by $\frac{5 t - 3}{t + 9}$ .

$\frac{(8 t - 3) (t - 5) (5 t - 3)}{(8 t - 3) (t + 9)}$

Step 7

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

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Step 7.1

Cancel the common factor.

$\frac{(8 t - 3) (t - 5) (5 t - 3)}{(8 t - 3) (t + 9)}$

Step 7.2

Rewrite the expression.

$\frac{(t - 5) (5 t - 3)}{t + 9}$