Trigonometry Examples

$3 x^{6} + 82 x^{3} + 27$

Step 1

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

$3 {(x^{3})}^{2} + 82 x^{3} + 27$

Step 2

Let $u = x^{3}$ . Substitute $u$ for all occurrences of $x^{3}$ .

$3 u^{2} + 82 u + 27$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 3 \cdot 27 = 81$ and whose sum is $b = 82$ .

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Step 3.1.1

Factor $82$ out of $82 u$ .

$3 u^{2} + 82 (u) + 27$

Step 3.1.2

Rewrite $82$ as $1$ plus $81$

$3 u^{2} + (1 + 81) u + 27$

Step 3.1.3

Apply the distributive property.

$3 u^{2} + 1 u + 81 u + 27$

Step 3.1.4

Multiply $u$ by $1$ .

$3 u^{2} + u + 81 u + 27$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(3 u^{2} + u) + 81 u + 27$

Step 3.2.2

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

$u (3 u + 1) + 27 (3 u + 1)$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

$(3 u + 1) (u + 27)$

Step 4

Replace all occurrences of $u$ with $x^{3}$ .

$(3 x^{3} + 1) (x^{3} + 27)$

Step 5

Rewrite $27$ as $3^{3}$ .

$(3 x^{3} + 1) (x^{3} + 3^{3})$

Step 6

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 3$ .

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

Step 7

Factor.

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Step 7.1

Simplify.

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Step 7.1.1

Multiply $3$ by $- 1$ .

$(3 x^{3} + 1) ((x + 3) (x^{2} - 3 x + 3^{2}))$

Step 7.1.2

Raise $3$ to the power of $2$ .

$(3 x^{3} + 1) ((x + 3) (x^{2} - 3 x + 9))$

Step 7.2

Remove unnecessary parentheses.

$(3 x^{3} + 1) (x + 3) (x^{2} - 3 x + 9)$