Trigonometry Examples

$p (x) = 3 x^{5} - 10 x^{4} - 21 x^{2} + 70 x$

Step 1

Set $3 x^{5} - 10 x^{4} - 21 x^{2} + 70 x$ equal to $0$ .

$3 x^{5} - 10 x^{4} - 21 x^{2} + 70 x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $x$ out of $3 x^{5} - 10 x^{4} - 21 x^{2} + 70 x$ .

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Step 2.1.1.1

Factor $x$ out of $3 x^{5}$ .

$x (3 x^{4}) - 10 x^{4} - 21 x^{2} + 70 x = 0$

Step 2.1.1.2

Factor $x$ out of $- 10 x^{4}$ .

$x (3 x^{4}) + x (- 10 x^{3}) - 21 x^{2} + 70 x = 0$

Step 2.1.1.3

Factor $x$ out of $- 21 x^{2}$ .

$x (3 x^{4}) + x (- 10 x^{3}) + x (- 21 x) + 70 x = 0$

Step 2.1.1.4

Factor $x$ out of $70 x$ .

$x (3 x^{4}) + x (- 10 x^{3}) + x (- 21 x) + x \cdot 70 = 0$

Step 2.1.1.5

Factor $x$ out of $x (3 x^{4}) + x (- 10 x^{3})$ .

$x (3 x^{4} - 10 x^{3}) + x (- 21 x) + x \cdot 70 = 0$

Step 2.1.1.6

Factor $x$ out of $x (3 x^{4} - 10 x^{3}) + x (- 21 x)$ .

$x (3 x^{4} - 10 x^{3} - 21 x) + x \cdot 70 = 0$

Step 2.1.1.7

Factor $x$ out of $x (3 x^{4} - 10 x^{3} - 21 x) + x \cdot 70$ .

$x (3 x^{4} - 10 x^{3} - 21 x + 70) = 0$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$x ((3 x^{4} - 10 x^{3}) - 21 x + 70) = 0$

Step 2.1.2.2

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

$x (x^{3} (3 x - 10) - 7 (3 x - 10)) = 0$

Step 2.1.3

Factor.

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Step 2.1.3.1

Factor the polynomial by factoring out the greatest common factor, $3 x - 10$ .

$x ((3 x - 10) (x^{3} - 7)) = 0$

Step 2.1.3.2

Remove unnecessary parentheses.

$x (3 x - 10) (x^{3} - 7) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$3 x - 10 = 0$

$x^{3} - 7 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $3 x - 10$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $3 x - 10$ equal to $0$ .

$3 x - 10 = 0$

Step 2.4.2

Solve $3 x - 10 = 0$ for $x$ .

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Step 2.4.2.1

Add $10$ to both sides of the equation.

$3 x = 10$

Step 2.4.2.2

Divide each term in $3 x = 10$ by $3$ and simplify.

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Step 2.4.2.2.1

Divide each term in $3 x = 10$ by $3$ .

$\frac{3 x}{3} = \frac{10}{3}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{10}{3}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{3}$

Step 2.5

Set $x^{3} - 7$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{3} - 7$ equal to $0$ .

$x^{3} - 7 = 0$

Step 2.5.2

Solve $x^{3} - 7 = 0$ for $x$ .

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Step 2.5.2.1

Add $7$ to both sides of the equation.

$x^{3} = 7$

Step 2.5.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{7}$

Step 2.6

The final solution is all the values that make $x (3 x - 10) (x^{3} - 7) = 0$ true.

$x = 0, \frac{10}{3}, \sqrt[3]{7}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{10}{3}, \sqrt[3]{7}$

Decimal Form:

$x = 0, 3.\overline{3}, 1.91293118 \dots$

Mixed Number Form:

$x = 0, 3 \frac{1}{3}, \sqrt[3]{7}$

Step 4