Algebra Examples

$\frac{t - 7}{t^{2} - 6 t - 16} \cdot \frac{t + 2}{t^{2} - 15 t + 56}$

Step 1

Factor $t^{2} - 6 t - 16$ 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 $- 16$ and whose sum is $- 6$ .

$- 8, 2$

Step 1.2

Write the factored form using these integers.

$\frac{t - 7}{(t - 8) (t + 2)} \cdot \frac{t + 2}{t^{2} - 15 t + 56}$

Step 2

Factor $t^{2} - 15 t + 56$ 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 $56$ and whose sum is $- 15$ .

$- 8, - 7$

Step 2.2

Write the factored form using these integers.

$\frac{t - 7}{(t - 8) (t + 2)} \cdot \frac{t + 2}{(t - 8) (t - 7)}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $t - 7$ .

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Step 3.1.1

Factor $t - 7$ out of $(t - 8) (t - 7)$ .

$\frac{t - 7}{(t - 8) (t + 2)} \cdot \frac{t + 2}{(t - 7) (t - 8)}$

Step 3.1.2

Cancel the common factor.

$\frac{t - 7}{(t - 8) (t + 2)} \cdot \frac{t + 2}{(t - 7) (t - 8)}$

Step 3.1.3

Rewrite the expression.

$\frac{1}{(t - 8) (t + 2)} \cdot \frac{t + 2}{t - 8}$

Step 3.2

Cancel the common factor of $t + 2$ .

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Step 3.2.1

Factor $t + 2$ out of $(t - 8) (t + 2)$ .

$\frac{1}{(t + 2) (t - 8)} \cdot \frac{t + 2}{t - 8}$

Step 3.2.2

Cancel the common factor.

$\frac{1}{(t + 2) (t - 8)} \cdot \frac{t + 2}{t - 8}$

Step 3.2.3

Rewrite the expression.

$\frac{1}{t - 8} \cdot \frac{1}{t - 8}$

Step 3.3

Multiply $\frac{1}{t - 8}$ by $\frac{1}{t - 8}$ .

$\frac{1}{(t - 8) (t - 8)}$

Step 4

Raise $t - 8$ to the power of $1$ .

$\frac{1}{{(t - 8)}^{1} (t - 8)}$

Step 5

Raise $t - 8$ to the power of $1$ .

$\frac{1}{{(t - 8)}^{1} {(t - 8)}^{1}}$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{{(t - 8)}^{1 + 1}}$

Step 7

Add $1$ and $1$ .

$\frac{1}{{(t - 8)}^{2}}$