Basic Math Examples

$\frac{3 z + 5 z - 12}{z^{2} + 10 z + 24} + \frac{z^{2} + 4 z + 7}{z^{2} + 10 z + 24} - \frac{2 z^{2} + 5 z + z}{z^{2} + 10 z + 24}$

Step 1

Combine the numerators over the common denominator.

$\frac{3 z + 5 z - 12 + z^{2} + 4 z + 7 - (2 z^{2} + 5 z + z)}{z^{2} + 10 z + 24}$

Step 2

Simplify each term.

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Step 2.1

Add $5 z$ and $z$ .

$\frac{3 z + 5 z - 12 + z^{2} + 4 z + 7 - (2 z^{2} + 6 z)}{z^{2} + 10 z + 24}$

Step 2.2

Apply the distributive property.

$\frac{3 z + 5 z - 12 + z^{2} + 4 z + 7 - (2 z^{2}) - (6 z)}{z^{2} + 10 z + 24}$

Step 2.3

Multiply $2$ by $- 1$ .

$\frac{3 z + 5 z - 12 + z^{2} + 4 z + 7 - 2 z^{2} - (6 z)}{z^{2} + 10 z + 24}$

Step 2.4

Multiply $6$ by $- 1$ .

$\frac{3 z + 5 z - 12 + z^{2} + 4 z + 7 - 2 z^{2} - 6 z}{z^{2} + 10 z + 24}$

Step 3

Simplify by adding terms.

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Step 3.1

Add $3 z$ and $5 z$ .

$\frac{8 z - 12 + z^{2} + 4 z + 7 - 2 z^{2} - 6 z}{z^{2} + 10 z + 24}$

Step 3.2

Add $8 z$ and $4 z$ .

$\frac{12 z - 12 + z^{2} + 7 - 2 z^{2} - 6 z}{z^{2} + 10 z + 24}$

Step 3.3

Subtract $6 z$ from $12 z$ .

$\frac{6 z - 12 + z^{2} + 7 - 2 z^{2}}{z^{2} + 10 z + 24}$

Step 3.4

Add $- 12$ and $7$ .

$\frac{6 z + z^{2} - 5 - 2 z^{2}}{z^{2} + 10 z + 24}$

Step 3.5

Subtract $2 z^{2}$ from $z^{2}$ .

$\frac{6 z - z^{2} - 5}{z^{2} + 10 z + 24}$

Step 3.6

Reorder $6 z$ and $- z^{2}$ .

$\frac{- z^{2} + 6 z - 5}{z^{2} + 10 z + 24}$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = - 1 \cdot - 5 = 5$ and whose sum is $b = 6$ .

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Step 4.1.1

Factor $6$ out of $6 z$ .

$\frac{- z^{2} + 6 (z) - 5}{z^{2} + 10 z + 24}$

Step 4.1.2

Rewrite $6$ as $1$ plus $5$

$\frac{- z^{2} + (1 + 5) z - 5}{z^{2} + 10 z + 24}$

Step 4.1.3

Apply the distributive property.

$\frac{- z^{2} + 1 z + 5 z - 5}{z^{2} + 10 z + 24}$

Step 4.1.4

Multiply $z$ by $1$ .

$\frac{- z^{2} + z + 5 z - 5}{z^{2} + 10 z + 24}$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$\frac{(- z^{2} + z) + 5 z - 5}{z^{2} + 10 z + 24}$

Step 4.2.2

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

$\frac{z (- z + 1) - 5 (- z + 1)}{z^{2} + 10 z + 24}$

Step 4.3

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

$\frac{(- z + 1) (z - 5)}{z^{2} + 10 z + 24}$

Step 5

Factor $z^{2} + 10 z + 24$ using the AC method.

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Step 5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $24$ and whose sum is $10$ .

$4, 6$

Step 5.2

Write the factored form using these integers.

$\frac{(- z + 1) (z - 5)}{(z + 4) (z + 6)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- z$ .

$\frac{(- (z) + 1) (z - 5)}{(z + 4) (z + 6)}$

Step 6.2

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{(- (z) - 1 (- 1)) (z - 5)}{(z + 4) (z + 6)}$

Step 6.3

Factor $- 1$ out of $- (z) - 1 (- 1)$ .

$\frac{- (z - 1) (z - 5)}{(z + 4) (z + 6)}$

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite $- (z - 1)$ as $- 1 (z - 1)$ .

$\frac{- 1 (z - 1) (z - 5)}{(z + 4) (z + 6)}$

Step 6.4.2

Move the negative in front of the fraction.

$- \frac{(z - 1) (z - 5)}{(z + 4) (z + 6)}$