Algebra Examples

$\frac{{(x - 1)}^{2}}{x^{2} - x - 12} \cdot \frac{x^{2} + x - 6}{x^{2} - 6 x + 5}$

Step 1

Factor $x^{2} - x - 12$ 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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 1.2

Write the factored form using these integers.

$\frac{{(x - 1)}^{2}}{(x - 4) (x + 3)} \cdot \frac{x^{2} + x - 6}{x^{2} - 6 x + 5}$

Step 2

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

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Step 2.1

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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 2.2

Write the factored form using these integers.

$\frac{{(x - 1)}^{2}}{(x - 4) (x + 3)} \cdot \frac{(x - 2) (x + 3)}{x^{2} - 6 x + 5}$

Step 3

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

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Step 3.1

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 $- 6$ .

$- 5, - 1$

Step 3.2

Write the factored form using these integers.

$\frac{{(x - 1)}^{2}}{(x - 4) (x + 3)} \cdot \frac{(x - 2) (x + 3)}{(x - 5) (x - 1)}$

Step 4

Simplify terms.

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Step 4.1

Combine.

$\frac{{(x - 1)}^{2} ((x - 2) (x + 3))}{(x - 4) (x + 3) ((x - 5) (x - 1))}$

Step 4.2

Cancel the common factor of ${(x - 1)}^{2}$ and $x - 1$ .

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Step 4.2.1

Factor $x - 1$ out of ${(x - 1)}^{2} ((x - 2) (x + 3))$ .

$\frac{(x - 1) ((x - 1) ((x - 2) (x + 3)))}{(x - 4) (x + 3) ((x - 5) (x - 1))}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $x - 1$ out of $(x - 4) (x + 3) ((x - 5) (x - 1))$ .

$\frac{(x - 1) ((x - 1) ((x - 2) (x + 3)))}{(x - 1) ((x - 4) (x + 3) (x - 5))}$

Step 4.2.2.2

Cancel the common factor.

$\frac{(x - 1) ((x - 1) ((x - 2) (x + 3)))}{(x - 1) ((x - 4) (x + 3) (x - 5))}$

Step 4.2.2.3

Rewrite the expression.

$\frac{(x - 1) ((x - 2) (x + 3))}{(x - 4) (x + 3) (x - 5)}$

Step 4.3

Cancel the common factor of $x + 3$ .

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Step 4.3.1

Cancel the common factor.

$\frac{(x - 1) ((x - 2) (x + 3))}{(x - 4) (x + 3) (x - 5)}$

Step 4.3.2

Rewrite the expression.

$\frac{(x - 1) (x - 2)}{(x - 4) (x - 5)}$