Simplify (187+6z-z^2)/(z^2-21z+68)

$\frac{187 + 6 z - z^{2}}{z^{2} - 21 z + 68}$

Simplify the numerator.

Rewrite $187$ as $196$ plus $- 9$

$\frac{196 - 9 + 6 z - z^{2}}{z^{2} - 21 z + 68}$

Factor using the perfect square rule.

Rewrite $9$ as $3^{2}$ .

$\frac{196 - (3^{2} - 6 z + z^{2})}{z^{2} - 21 z + 68}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 z = 2 \cdot 3 \cdot z$

Rewrite the polynomial.

$\frac{196 - (3^{2} - 2 \cdot 3 \cdot z + z^{2})}{z^{2} - 21 z + 68}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 3$ and $b = z$ .

$\frac{196 - {(3 - z)}^{2}}{z^{2} - 21 z + 68}$

Rewrite $196$ as $14^{2}$ .

$\frac{14^{2} - {(3 - z)}^{2}}{z^{2} - 21 z + 68}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 14$ and $b = 3 - z$ .

$\frac{(14 + 3 - z) (14 - (3 - z))}{z^{2} - 21 z + 68}$

Simplify.

Add $14$ and $3$ .

$\frac{(17 - z) (14 - (3 - z))}{z^{2} - 21 z + 68}$

Apply the distributive property.

$\frac{(17 - z) (14 - 1 \cdot 3 - - z)}{z^{2} - 21 z + 68}$

Multiply $- 1$ by $3$ .

$\frac{(17 - z) (14 - 3 - - z)}{z^{2} - 21 z + 68}$

Multiply $- - z$ .

Multiply $- 1$ by $- 1$ .

$\frac{(17 - z) (14 - 3 + 1 z)}{z^{2} - 21 z + 68}$

Multiply $z$ by $1$ .

$\frac{(17 - z) (14 - 3 + z)}{z^{2} - 21 z + 68}$

Subtract $3$ from $14$ .

$\frac{(17 - z) (11 + z)}{z^{2} - 21 z + 68}$

Factor $z^{2} - 21 z + 68$ using the AC method.

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 $68$ and whose sum is $- 21$ .

$- 17, - 4$

Write the factored form using these integers.

$\frac{(17 - z) (11 + z)}{(z - 17) (z - 4)}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $17 - z$ and $z - 17$ .

Rewrite $17$ as $- 1 (- 17)$ .

$\frac{(- 1 (- 17) - z) (11 + z)}{(z - 17) (z - 4)}$

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

$\frac{(- 1 (- 17) - (z)) (11 + z)}{(z - 17) (z - 4)}$

Factor $- 1$ out of $- 1 (- 17) - (z)$ .

$\frac{- 1 (- 17 + z) (11 + z)}{(z - 17) (z - 4)}$

Reorder terms.

$\frac{- 1 (z - 17) (11 + z)}{(z - 17) (z - 4)}$

Cancel the common factor.

$\frac{- 1 (z - 17) (11 + z)}{(z - 17) (z - 4)}$

Rewrite the expression.

$\frac{- 1 (11 + z)}{z - 4}$

Move the negative in front of the fraction.

$- \frac{11 + z}{z - 4}$

Simplify (187+6z-z^2)/(z^2-21z+68)

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