Find the Product (z-5/3)(z-2/3)

$(z - \frac{5}{3}) (z - \frac{2}{3})$

Expand $(z - \frac{5}{3}) (z - \frac{2}{3})$ using the FOIL Method.

Apply the distributive property.

$z (z - \frac{2}{3}) - \frac{5}{3} (z - \frac{2}{3})$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $z$ by $z$ .

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

Combine $z$ and $\frac{2}{3}$ .

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

Move $2$ to the left of $z$ .

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

Combine $z$ and $\frac{5}{3}$ .

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

Move $5$ to the left of $z$ .

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

Multiply $- \frac{5}{3} (- \frac{2}{3})$ .

Multiply $- 1$ by $- 1$ .

$z^{2} - \frac{2 z}{3} - \frac{5 z}{3} + 1 (\frac{5}{3}) \frac{2}{3}$

Multiply $\frac{5}{3}$ by $1$ .

$z^{2} - \frac{2 z}{3} - \frac{5 z}{3} + \frac{5}{3} \cdot \frac{2}{3}$

Multiply $\frac{5}{3}$ by $\frac{2}{3}$ .

$z^{2} - \frac{2 z}{3} - \frac{5 z}{3} + \frac{5 \cdot 2}{3 \cdot 3}$

Multiply $5$ by $2$ .

$z^{2} - \frac{2 z}{3} - \frac{5 z}{3} + \frac{10}{3 \cdot 3}$

Multiply $3$ by $3$ .

$z^{2} - \frac{2 z}{3} - \frac{5 z}{3} + \frac{10}{9}$

Combine the numerators over the common denominator.

$z^{2} + \frac{- 2 z - 5 z}{3} + \frac{10}{9}$

Subtract $5 z$ from $- 2 z$ .

$z^{2} + \frac{- 7 z}{3} + \frac{10}{9}$

Move the negative in front of the fraction.

$z^{2} - \frac{7 z}{3} + \frac{10}{9}$