Basic Math Examples

$\frac{5}{z - 4} + \frac{4 z}{z - 3}$

Step 1

To write $\frac{5}{z - 4}$ as a fraction with a common denominator, multiply by $\frac{z - 3}{z - 3}$ .

$\frac{5}{z - 4} \cdot \frac{z - 3}{z - 3} + \frac{4 z}{z - 3}$

Step 2

To write $\frac{4 z}{z - 3}$ as a fraction with a common denominator, multiply by $\frac{z - 4}{z - 4}$ .

$\frac{5}{z - 4} \cdot \frac{z - 3}{z - 3} + \frac{4 z}{z - 3} \cdot \frac{z - 4}{z - 4}$

Step 3

Write each expression with a common denominator of $(z - 4) (z - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5}{z - 4}$ by $\frac{z - 3}{z - 3}$ .

$\frac{5 (z - 3)}{(z - 4) (z - 3)} + \frac{4 z}{z - 3} \cdot \frac{z - 4}{z - 4}$

Step 3.2

Multiply $\frac{4 z}{z - 3}$ by $\frac{z - 4}{z - 4}$ .

$\frac{5 (z - 3)}{(z - 4) (z - 3)} + \frac{4 z (z - 4)}{(z - 3) (z - 4)}$

Step 3.3

Reorder the factors of $(z - 4) (z - 3)$ .

$\frac{5 (z - 3)}{(z - 3) (z - 4)} + \frac{4 z (z - 4)}{(z - 3) (z - 4)}$

Step 4

Combine the numerators over the common denominator.

$\frac{5 (z - 3) + 4 z (z - 4)}{(z - 3) (z - 4)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{5 z + 5 \cdot - 3 + 4 z (z - 4)}{(z - 3) (z - 4)}$

Step 5.2

Multiply $5$ by $- 3$ .

$\frac{5 z - 15 + 4 z (z - 4)}{(z - 3) (z - 4)}$

Step 5.3

Apply the distributive property.

$\frac{5 z - 15 + 4 z \cdot z + 4 z \cdot - 4}{(z - 3) (z - 4)}$

Step 5.4

Multiply $z$ by $z$ by adding the exponents.

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Step 5.4.1

Move $z$ .

$\frac{5 z - 15 + 4 (z \cdot z) + 4 z \cdot - 4}{(z - 3) (z - 4)}$

Step 5.4.2

Multiply $z$ by $z$ .

$\frac{5 z - 15 + 4 z^{2} + 4 z \cdot - 4}{(z - 3) (z - 4)}$

Step 5.5

Multiply $- 4$ by $4$ .

$\frac{5 z - 15 + 4 z^{2} - 16 z}{(z - 3) (z - 4)}$

Step 5.6

Subtract $16 z$ from $5 z$ .

$\frac{- 11 z - 15 + 4 z^{2}}{(z - 3) (z - 4)}$

Step 5.7

Reorder terms.

$\frac{4 z^{2} - 11 z - 15}{(z - 3) (z - 4)}$

Step 5.8

Factor by grouping.

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Step 5.8.1

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 $a \cdot c = 4 \cdot - 15 = - 60$ and whose sum is $b = - 11$ .

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Step 5.8.1.1

Factor $- 11$ out of $- 11 z$ .

$\frac{4 z^{2} - 11 (z) - 15}{(z - 3) (z - 4)}$

Step 5.8.1.2

Rewrite $- 11$ as $4$ plus $- 15$

$\frac{4 z^{2} + (4 - 15) z - 15}{(z - 3) (z - 4)}$

Step 5.8.1.3

Apply the distributive property.

$\frac{4 z^{2} + 4 z - 15 z - 15}{(z - 3) (z - 4)}$

Step 5.8.2

Factor out the greatest common factor from each group.

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Step 5.8.2.1

Group the first two terms and the last two terms.

$\frac{(4 z^{2} + 4 z) - 15 z - 15}{(z - 3) (z - 4)}$

Step 5.8.2.2

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

$\frac{4 z (z + 1) - 15 (z + 1)}{(z - 3) (z - 4)}$

Step 5.8.3

Factor the polynomial by factoring out the greatest common factor, $z + 1$ .

$\frac{(z + 1) (4 z - 15)}{(z - 3) (z - 4)}$