Finite Math Examples

$\frac{x^{2} + 6 x + 5}{2 x^{2} + 7 x + 5}$

Step 1

Factor $x^{2} + 6 x + 5$ using the AC method.

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Step 1.1

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 $5$ and whose sum is $6$ .

$1, 5$

Step 1.2

Write the factored form using these integers.

$\frac{(x + 1) (x + 5)}{2 x^{2} + 7 x + 5}$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 2 \cdot 5 = 10$ and whose sum is $b = 7$ .

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Step 2.1.1

Factor $7$ out of $7 x$ .

$\frac{(x + 1) (x + 5)}{2 x^{2} + 7 (x) + 5}$

Step 2.1.2

Rewrite $7$ as $2$ plus $5$

$\frac{(x + 1) (x + 5)}{2 x^{2} + (2 + 5) x + 5}$

Step 2.1.3

Apply the distributive property.

$\frac{(x + 1) (x + 5)}{2 x^{2} + 2 x + 5 x + 5}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(x + 1) (x + 5)}{(2 x^{2} + 2 x) + 5 x + 5}$

Step 2.2.2

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

$\frac{(x + 1) (x + 5)}{2 x (x + 1) + 5 (x + 1)}$

Step 2.3

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

$\frac{(x + 1) (x + 5)}{(x + 1) (2 x + 5)}$

Step 3

Cancel the common factor of $x + 1$ .

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Step 3.1

Cancel the common factor.

$\frac{(x + 1) (x + 5)}{(x + 1) (2 x + 5)}$

Step 3.2

Rewrite the expression.

$\frac{x + 5}{2 x + 5}$