Trigonometry Examples

$\frac{t^{4} + 6 t^{3} - 7 t^{2}}{t^{6} + 2 t^{5} - 3 t^{4}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $t^{2}$ out of $t^{4} + 6 t^{3} - 7 t^{2}$ .

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Step 1.1.1

Factor $t^{2}$ out of $t^{4}$ .

$\frac{t^{2} t^{2} + 6 t^{3} - 7 t^{2}}{t^{6} + 2 t^{5} - 3 t^{4}}$

Step 1.1.2

Factor $t^{2}$ out of $6 t^{3}$ .

$\frac{t^{2} t^{2} + t^{2} (6 t) - 7 t^{2}}{t^{6} + 2 t^{5} - 3 t^{4}}$

Step 1.1.3

Factor $t^{2}$ out of $- 7 t^{2}$ .

$\frac{t^{2} t^{2} + t^{2} (6 t) + t^{2} \cdot - 7}{t^{6} + 2 t^{5} - 3 t^{4}}$

Step 1.1.4

Factor $t^{2}$ out of $t^{2} t^{2} + t^{2} (6 t)$ .

$\frac{t^{2} (t^{2} + 6 t) + t^{2} \cdot - 7}{t^{6} + 2 t^{5} - 3 t^{4}}$

Step 1.1.5

Factor $t^{2}$ out of $t^{2} (t^{2} + 6 t) + t^{2} \cdot - 7$ .

$\frac{t^{2} (t^{2} + 6 t - 7)}{t^{6} + 2 t^{5} - 3 t^{4}}$

Step 1.2

Factor $t^{2} + 6 t - 7$ using the AC method.

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Step 1.2.1

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 $- 7$ and whose sum is $6$ .

$- 1, 7$

Step 1.2.2

Write the factored form using these integers.

$\frac{t^{2} ((t - 1) (t + 7))}{t^{6} + 2 t^{5} - 3 t^{4}}$

$\frac{t^{2} (t - 1) (t + 7)}{t^{6} + 2 t^{5} - 3 t^{4}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $t^{4}$ out of $t^{6} + 2 t^{5} - 3 t^{4}$ .

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Step 2.1.1

Factor $t^{4}$ out of $t^{6}$ .

$\frac{t^{2} (t - 1) (t + 7)}{t^{4} t^{2} + 2 t^{5} - 3 t^{4}}$

Step 2.1.2

Factor $t^{4}$ out of $2 t^{5}$ .

$\frac{t^{2} (t - 1) (t + 7)}{t^{4} t^{2} + t^{4} (2 t) - 3 t^{4}}$

Step 2.1.3

Factor $t^{4}$ out of $- 3 t^{4}$ .

$\frac{t^{2} (t - 1) (t + 7)}{t^{4} t^{2} + t^{4} (2 t) + t^{4} \cdot - 3}$

Step 2.1.4

Factor $t^{4}$ out of $t^{4} t^{2} + t^{4} (2 t)$ .

$\frac{t^{2} (t - 1) (t + 7)}{t^{4} (t^{2} + 2 t) + t^{4} \cdot - 3}$

Step 2.1.5

Factor $t^{4}$ out of $t^{4} (t^{2} + 2 t) + t^{4} \cdot - 3$ .

$\frac{t^{2} (t - 1) (t + 7)}{t^{4} (t^{2} + 2 t - 3)}$

Step 2.2

Factor $t^{2} + 2 t - 3$ using the AC method.

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Step 2.2.1

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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 2.2.2

Write the factored form using these integers.

$\frac{t^{2} (t - 1) (t + 7)}{t^{4} ((t - 1) (t + 3))}$

$\frac{t^{2} (t - 1) (t + 7)}{t^{4} (t - 1) (t + 3)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $t^{2}$ and $t^{4}$ .

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Step 3.1.1

Factor $t^{2}$ out of $t^{2} (t - 1) (t + 7)$ .

$\frac{t^{2} ((t - 1) (t + 7))}{t^{4} (t - 1) (t + 3)}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $t^{2}$ out of $t^{4} (t - 1) (t + 3)$ .

$\frac{t^{2} ((t - 1) (t + 7))}{t^{2} ((t^{2} (t - 1)) (t + 3))}$

Step 3.1.2.2

Cancel the common factor.

$\frac{t^{2} ((t - 1) (t + 7))}{t^{2} ((t^{2} (t - 1)) (t + 3))}$

Step 3.1.2.3

Rewrite the expression.

$\frac{(t - 1) (t + 7)}{(t^{2} (t - 1)) (t + 3)}$

Step 3.2

Cancel the common factor of $t - 1$ .

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Step 3.2.1

Cancel the common factor.

$\frac{(t - 1) (t + 7)}{t^{2} (t - 1) (t + 3)}$

Step 3.2.2

Rewrite the expression.

$\frac{t + 7}{(t^{2}) (t + 3)}$

$\frac{t + 7}{t^{2} (t + 3)}$