Basic Math Examples

$(z - \frac{7}{3}) (z + \frac{3}{7})$

Step 1

Expand $(z - \frac{7}{3}) (z + \frac{3}{7})$ using the FOIL Method.

Tap for more steps...

Step 1.1

Apply the distributive property.

$z (z + \frac{3}{7}) - \frac{7}{3} (z + \frac{3}{7})$

Step 1.2

Apply the distributive property.

$z \cdot z + z \frac{3}{7} - \frac{7}{3} (z + \frac{3}{7})$

Step 1.3

Apply the distributive property.

$z \cdot z + z \frac{3}{7} - \frac{7}{3} z - \frac{7}{3} \cdot \frac{3}{7}$

Step 2

Simplify and combine like terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Multiply $z$ by $z$ .

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

Step 2.1.2

Combine $z$ and $\frac{3}{7}$ .

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

Step 2.1.3

Move $3$ to the left of $z$ .

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

Step 2.1.4

Combine $z$ and $\frac{7}{3}$ .

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

Step 2.1.5

Move $7$ to the left of $z$ .

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

Step 2.1.6

Cancel the common factor of $7$ .

Tap for more steps...

Step 2.1.6.1

Move the leading negative in $- \frac{7}{3}$ into the numerator.

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

Step 2.1.6.2

Factor $7$ out of $- 7$ .

$z^{2} + \frac{3 z}{7} - \frac{7 z}{3} + \frac{7 (- 1)}{3} \cdot \frac{3}{7}$

Step 2.1.6.3

Cancel the common factor.

$z^{2} + \frac{3 z}{7} - \frac{7 z}{3} + \frac{7 \cdot - 1}{3} \cdot \frac{3}{7}$

Step 2.1.6.4

Rewrite the expression.

$z^{2} + \frac{3 z}{7} - \frac{7 z}{3} + \frac{- 1}{3} \cdot 3$

Step 2.1.7

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.1.7.1

Cancel the common factor.

$z^{2} + \frac{3 z}{7} - \frac{7 z}{3} + \frac{- 1}{3} \cdot 3$

Step 2.1.7.2

Rewrite the expression.

$z^{2} + \frac{3 z}{7} - \frac{7 z}{3} - 1$

Step 2.2

To write $\frac{3 z}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$z^{2} + \frac{3 z}{7} \cdot \frac{3}{3} - \frac{7 z}{3} - 1$

Step 2.3

To write $- \frac{7 z}{3}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$z^{2} + \frac{3 z}{7} \cdot \frac{3}{3} - \frac{7 z}{3} \cdot \frac{7}{7} - 1$

Step 2.4

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.4.1

Multiply $\frac{3 z}{7}$ by $\frac{3}{3}$ .

$z^{2} + \frac{3 z \cdot 3}{7 \cdot 3} - \frac{7 z}{3} \cdot \frac{7}{7} - 1$

Step 2.4.2

Multiply $7$ by $3$ .

$z^{2} + \frac{3 z \cdot 3}{21} - \frac{7 z}{3} \cdot \frac{7}{7} - 1$

Step 2.4.3

Multiply $\frac{7 z}{3}$ by $\frac{7}{7}$ .

$z^{2} + \frac{3 z \cdot 3}{21} - \frac{7 z \cdot 7}{3 \cdot 7} - 1$

Step 2.4.4

Multiply $3$ by $7$ .

$z^{2} + \frac{3 z \cdot 3}{21} - \frac{7 z \cdot 7}{21} - 1$

Step 2.5

Combine the numerators over the common denominator.

$z^{2} + \frac{3 z \cdot 3 - 7 z \cdot 7}{21} - 1$

Step 2.6

To write $z^{2}$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

$z^{2} \cdot \frac{21}{21} + \frac{3 z \cdot 3 - 7 z \cdot 7}{21} - 1$

Step 2.7

Combine $z^{2}$ and $\frac{21}{21}$ .

$\frac{z^{2} \cdot 21}{21} + \frac{3 z \cdot 3 - 7 z \cdot 7}{21} - 1$

Step 2.8

Combine the numerators over the common denominator.

$\frac{z^{2} \cdot 21 + 3 z \cdot 3 - 7 z \cdot 7}{21} - 1$

Step 2.9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

$\frac{z^{2} \cdot 21 + 3 z \cdot 3 - 7 z \cdot 7}{21} - 1 \cdot \frac{21}{21}$

Step 2.10

Combine $- 1$ and $\frac{21}{21}$ .

$\frac{z^{2} \cdot 21 + 3 z \cdot 3 - 7 z \cdot 7}{21} + \frac{- 1 \cdot 21}{21}$

Step 2.11

Combine the numerators over the common denominator.

$\frac{z^{2} \cdot 21 + 3 z \cdot 3 - 7 z \cdot 7 - 1 \cdot 21}{21}$

Step 3

Simplify the numerator.

Tap for more steps...

Step 3.1

Move $21$ to the left of $z^{2}$ .

$\frac{21 \cdot z^{2} + 3 z \cdot 3 - 7 z \cdot 7 - 1 \cdot 21}{21}$

Step 3.2

Multiply $3$ by $3$ .

$\frac{21 z^{2} + 9 z - 7 z \cdot 7 - 1 \cdot 21}{21}$

Step 3.3

Multiply $7$ by $- 7$ .

$\frac{21 z^{2} + 9 z - 49 z - 1 \cdot 21}{21}$

Step 3.4

Multiply $- 1$ by $21$ .

$\frac{21 z^{2} + 9 z - 49 z - 21}{21}$

Step 3.5

Subtract $49 z$ from $9 z$ .

$\frac{21 z^{2} - 40 z - 21}{21}$

Step 3.6

Factor by grouping.

Tap for more steps...

Step 3.6.1

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 $a \cdot c = 21 \cdot - 21 = - 441$ and whose sum is $b = - 40$ .

Tap for more steps...

Step 3.6.1.1

Factor $- 40$ out of $- 40 z$ .

$\frac{21 z^{2} - 40 (z) - 21}{21}$

Step 3.6.1.2

Rewrite $- 40$ as $9$ plus $- 49$

$\frac{21 z^{2} + (9 - 49) z - 21}{21}$

Step 3.6.1.3

Apply the distributive property.

$\frac{21 z^{2} + 9 z - 49 z - 21}{21}$

Step 3.6.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.6.2.1

Group the first two terms and the last two terms.

$\frac{(21 z^{2} + 9 z) - 49 z - 21}{21}$

Step 3.6.2.2

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

$\frac{3 z (7 z + 3) - 7 (7 z + 3)}{21}$

Step 3.6.3

Factor the polynomial by factoring out the greatest common factor, $7 z + 3$ .

$\frac{(7 z + 3) (3 z - 7)}{21}$