Algebra Examples

$\frac{24 + 5 n - n^{2}}{n^{2} - 16 n + 64}$

Step 1

Factor by grouping.

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Step 1.1

Reorder terms.

$\frac{- n^{2} + 5 n + 24}{n^{2} - 16 n + 64}$

Step 1.2

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 $a \cdot c = - 1 \cdot 24 = - 24$ and whose sum is $b = 5$ .

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Step 1.2.1

Factor $5$ out of $5 n$ .

$\frac{- n^{2} + 5 (n) + 24}{n^{2} - 16 n + 64}$

Step 1.2.2

Rewrite $5$ as $- 3$ plus $8$

$\frac{- n^{2} + (- 3 + 8) n + 24}{n^{2} - 16 n + 64}$

Step 1.2.3

Apply the distributive property.

$\frac{- n^{2} - 3 n + 8 n + 24}{n^{2} - 16 n + 64}$

Step 1.3

Factor out the greatest common factor from each group.

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Step 1.3.1

Group the first two terms and the last two terms.

$\frac{(- n^{2} - 3 n) + 8 n + 24}{n^{2} - 16 n + 64}$

Step 1.3.2

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

$\frac{n (- n - 3) - 8 (- n - 3)}{n^{2} - 16 n + 64}$

Step 1.4

Factor the polynomial by factoring out the greatest common factor, $- n - 3$ .

$\frac{(- n - 3) (n - 8)}{n^{2} - 16 n + 64}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $64$ as $8^{2}$ .

$\frac{(- n - 3) (n - 8)}{n^{2} - 16 n + 8^{2}}$

Step 2.2

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

$16 n = 2 \cdot n \cdot 8$

Step 2.3

Rewrite the polynomial.

$\frac{(- n - 3) (n - 8)}{n^{2} - 2 \cdot n \cdot 8 + 8^{2}}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = n$ and $b = 8$ .

$\frac{(- n - 3) (n - 8)}{{(n - 8)}^{2}}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $n - 8$ and ${(n - 8)}^{2}$ .

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Step 3.1.1

Factor $n - 8$ out of $(- n - 3) (n - 8)$ .

$\frac{(n - 8) (- n - 3)}{{(n - 8)}^{2}}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $n - 8$ out of ${(n - 8)}^{2}$ .

$\frac{(n - 8) (- n - 3)}{(n - 8) (n - 8)}$

Step 3.1.2.2

Cancel the common factor.

$\frac{(n - 8) (- n - 3)}{(n - 8) (n - 8)}$

Step 3.1.2.3

Rewrite the expression.

$\frac{- n - 3}{n - 8}$

Step 3.2

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

$\frac{- (n) - 3}{n - 8}$

Step 3.3

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{- (n) - 1 (3)}{n - 8}$

Step 3.4

Factor $- 1$ out of $- (n) - 1 (3)$ .

$\frac{- (n + 3)}{n - 8}$

Step 3.5

Simplify the expression.

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Step 3.5.1

Rewrite $- (n + 3)$ as $- 1 (n + 3)$ .

$\frac{- 1 (n + 3)}{n - 8}$

Step 3.5.2

Move the negative in front of the fraction.

$- \frac{n + 3}{n - 8}$