Simplify (-n^2+16n-63)/(n^2-2n-35)

$\frac{- n^{2} + 16 n - 63}{n^{2} - 2 n - 35}$

Factor by grouping.

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 $a \cdot c = - 1 \cdot - 63 = 63$ and whose sum is $b = 16$ .

Factor $16$ out of $16 n$ .

$\frac{- n^{2} + 16 (n) - 63}{n^{2} - 2 n - 35}$

Rewrite $16$ as $7$ plus $9$

$\frac{- n^{2} + (7 + 9) n - 63}{n^{2} - 2 n - 35}$

Apply the distributive property.

$\frac{- n^{2} + 7 n + 9 n - 63}{n^{2} - 2 n - 35}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- n^{2} + 7 n) + 9 n - 63}{n^{2} - 2 n - 35}$

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

$\frac{n (- n + 7) - 9 (- n + 7)}{n^{2} - 2 n - 35}$

Factor the polynomial by factoring out the greatest common factor, $- n + 7$ .

$\frac{(- n + 7) (n - 9)}{n^{2} - 2 n - 35}$

Factor $n^{2} - 2 n - 35$ 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 $- 35$ and whose sum is $- 2$ .

$- 7, 5$

Write the factored form using these integers.

$\frac{(- n + 7) (n - 9)}{(n - 7) (n + 5)}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- n + 7$ and $n - 7$ .

Factor $- 1$ out of $- n$ .

$\frac{(- (n) + 7) (n - 9)}{(n - 7) (n + 5)}$

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{(- (n) - 1 (- 7)) (n - 9)}{(n - 7) (n + 5)}$

Factor $- 1$ out of $- (n) - 1 (- 7)$ .

$\frac{- (n - 7) (n - 9)}{(n - 7) (n + 5)}$

Rewrite $- (n - 7)$ as $- 1 (n - 7)$ .

$\frac{- 1 (n - 7) (n - 9)}{(n - 7) (n + 5)}$

Cancel the common factor.

$\frac{- 1 (n - 7) (n - 9)}{(n - 7) (n + 5)}$

Rewrite the expression.

$\frac{- 1 (n - 9)}{n + 5}$

Move the negative in front of the fraction.

$- \frac{n - 9}{n + 5}$

Simplify (-n^2+16n-63)/(n^2-2n-35)

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