Algebra Examples

$\frac{t^{2} - 2 t - 3}{t + 1} \cdot \frac{3 t - 3}{t^{2} + 2 t - 3}$

Step 1

Factor $t^{2} - 2 t - 3$ 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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 1.2

Write the factored form using these integers.

$\frac{(t - 3) (t + 1)}{t + 1} \cdot \frac{3 t - 3}{t^{2} + 2 t - 3}$

Step 2

Factor $3$ out of $3 t - 3$ .

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Step 2.1

Factor $3$ out of $3 t$ .

$\frac{(t - 3) (t + 1)}{t + 1} \cdot \frac{3 (t) - 3}{t^{2} + 2 t - 3}$

Step 2.2

Factor $3$ out of $- 3$ .

$\frac{(t - 3) (t + 1)}{t + 1} \cdot \frac{3 (t) + 3 (- 1)}{t^{2} + 2 t - 3}$

Step 2.3

Factor $3$ out of $3 (t) + 3 (- 1)$ .

$\frac{(t - 3) (t + 1)}{t + 1} \cdot \frac{3 (t - 1)}{t^{2} + 2 t - 3}$

Step 3

Factor $t^{2} + 2 t - 3$ 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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 3.2

Write the factored form using these integers.

$\frac{(t - 3) (t + 1)}{t + 1} \cdot \frac{3 (t - 1)}{(t - 1) (t + 3)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $t + 1$ .

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Step 4.1.1

Cancel the common factor.

$\frac{(t - 3) (t + 1)}{t + 1} \cdot \frac{3 (t - 1)}{(t - 1) (t + 3)}$

Step 4.1.2

Divide $t - 3$ by $1$ .

$(t - 3) \cdot \frac{3 (t - 1)}{(t - 1) (t + 3)}$

Step 4.2

Cancel the common factor of $t - 1$ .

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Step 4.2.1

Cancel the common factor.

$(t - 3) \cdot \frac{3 (t - 1)}{(t - 1) (t + 3)}$

Step 4.2.2

Rewrite the expression.

$(t - 3) \cdot \frac{3}{t + 3}$

Step 4.3

Multiply $t - 3$ by $\frac{3}{t + 3}$ .

$\frac{(t - 3) \cdot 3}{t + 3}$

Step 4.4

Move $3$ to the left of $t - 3$ .

$\frac{3 (t - 3)}{t + 3}$