Precalculus Examples

$\frac{5 x + 41 x + 42}{5 x} \cdot \frac{3 x - 5 + 18}{15 x^{2} - 27 x - 54}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Add $5 x$ and $41 x$ .

$\frac{46 x + 42}{5 x} \cdot \frac{3 x - 5 + 18}{15 x^{2} - 27 x - 54}$

Step 1.2

Factor $2$ out of $46 x + 42$ .

Tap for more steps...

Step 1.2.1

Factor $2$ out of $46 x$ .

$\frac{2 (23 x) + 42}{5 x} \cdot \frac{3 x - 5 + 18}{15 x^{2} - 27 x - 54}$

Step 1.2.2

Factor $2$ out of $42$ .

$\frac{2 (23 x) + 2 (21)}{5 x} \cdot \frac{3 x - 5 + 18}{15 x^{2} - 27 x - 54}$

Step 1.2.3

Factor $2$ out of $2 (23 x) + 2 (21)$ .

$\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x - 5 + 18}{15 x^{2} - 27 x - 54}$

Step 2

Add $- 5$ and $18$ .

$\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x + 13}{15 x^{2} - 27 x - 54}$

Step 3

Simplify the denominator.

Tap for more steps...

Step 3.1

Factor $3$ out of $15 x^{2} - 27 x - 54$ .

Tap for more steps...

Step 3.1.1

Factor $3$ out of $15 x^{2}$ .

$\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x + 13}{3 (5 x^{2}) - 27 x - 54}$

Step 3.1.2

Factor $3$ out of $- 27 x$ .

$\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x + 13}{3 (5 x^{2}) + 3 (- 9 x) - 54}$

Step 3.1.3

Factor $3$ out of $- 54$ .

$\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x + 13}{3 (5 x^{2}) + 3 (- 9 x) + 3 (- 18)}$

Step 3.1.4

Factor $3$ out of $3 (5 x^{2}) + 3 (- 9 x)$ .

$\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x + 13}{3 (5 x^{2} - 9 x) + 3 (- 18)}$

Step 3.1.5

Factor $3$ out of $3 (5 x^{2} - 9 x) + 3 (- 18)$ .

$\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x + 13}{3 (5 x^{2} - 9 x - 18)}$

Step 3.2

Factor by grouping.

Tap for more steps...

Step 3.2.1

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 $a \cdot c = 5 \cdot - 18 = - 90$ and whose sum is $b = - 9$ .

Tap for more steps...

Step 3.2.1.1

Factor $- 9$ out of $- 9 x$ .

$\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x + 13}{3 (5 x^{2} - 9 (x) - 18)}$

Step 3.2.1.2

Rewrite $- 9$ as $6$ plus $- 15$

$\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x + 13}{3 (5 x^{2} + (6 - 15) x - 18)}$

Step 3.2.1.3

Apply the distributive property.

$\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x + 13}{3 (5 x^{2} + 6 x - 15 x - 18)}$

Step 3.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.2.2.1

Group the first two terms and the last two terms.

$\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x + 13}{3 ((5 x^{2} + 6 x) - 15 x - 18)}$

Step 3.2.2.2

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

$\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x + 13}{3 (x (5 x + 6) - 3 (5 x + 6))}$

Step 3.2.3

Factor the polynomial by factoring out the greatest common factor, $5 x + 6$ .

$\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x + 13}{3 ((5 x + 6) (x - 3))}$

$\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x + 13}{3 (5 x + 6) (x - 3)}$

Step 4

Multiply $\frac{2 (23 x + 21)}{5 x} \cdot \frac{3 x + 13}{3 (5 x + 6) (x - 3)}$ .

Tap for more steps...

Step 4.1

Multiply $\frac{2 (23 x + 21)}{5 x}$ by $\frac{3 x + 13}{3 (5 x + 6) (x - 3)}$ .

$\frac{2 (23 x + 21) (3 x + 13)}{5 x (3 (5 x + 6) (x - 3))}$

Step 4.2

Multiply $3$ by $5$ .

$\frac{2 (23 x + 21) (3 x + 13)}{15 x ((5 x + 6) (x - 3))}$

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

Rewrite.

$\frac{2 (23 x + 21) (3 x + 13)}{5 x (3 (5 x + 6) (x - 3))}$

Step 5.2

Multiply $3$ by $5$ .

$\frac{2 (23 x + 21) (3 x + 13)}{15 x ((5 x + 6) (x - 3))}$

Step 5.3

Remove unnecessary parentheses.

$\frac{2 (23 x + 21) (3 x + 13)}{15 x (5 x + 6) (x - 3)}$