Algebra Examples

$f (x) = 3 x^{3} - 10 x^{2} + 3 x$

Step 1

Set $3 x^{3} - 10 x^{2} + 3 x$ equal to $0$ .

$3 x^{3} - 10 x^{2} + 3 x = 0$

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 $3 x^{3} - 10 x^{2} + 3 x$ .

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Step 2.1.1.1

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

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

Step 2.1.1.2

Factor $x$ out of $- 10 x^{2}$ .

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

Step 2.1.1.3

Factor $x$ out of $3 x$ .

$x (3 x^{2}) + x (- 10 x) + x \cdot 3 = 0$

Step 2.1.1.4

Factor $x$ out of $x (3 x^{2}) + x (- 10 x)$ .

$x (3 x^{2} - 10 x) + x \cdot 3 = 0$

Step 2.1.1.5

Factor $x$ out of $x (3 x^{2} - 10 x) + x \cdot 3$ .

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

Step 2.1.2

Factor.

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Step 2.1.2.1

Factor by grouping.

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Step 2.1.2.1.1

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 $a \cdot c = 3 \cdot 3 = 9$ and whose sum is $b = - 10$ .

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Step 2.1.2.1.1.1

Factor $- 10$ out of $- 10 x$ .

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

Step 2.1.2.1.1.2

Rewrite $- 10$ as $- 1$ plus $- 9$

$x (3 x^{2} + (- 1 - 9) x + 3) = 0$

Step 2.1.2.1.1.3

Apply the distributive property.

$x (3 x^{2} - 1 x - 9 x + 3) = 0$

Step 2.1.2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1.2.1

Group the first two terms and the last two terms.

$x ((3 x^{2} - 1 x) - 9 x + 3) = 0$

Step 2.1.2.1.2.2

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

$x (x (3 x - 1) - 3 (3 x - 1)) = 0$

Step 2.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$x ((3 x - 1) (x - 3)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$x (3 x - 1) (x - 3) = 0$

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 $0$ .

$x = 0$

$3 x - 1 = 0$

$x - 3 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $3 x - 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 2.4.2

Solve $3 x - 1 = 0$ for $x$ .

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Step 2.4.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 2.4.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 2.4.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

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.

$\frac{3 x}{3} = \frac{1}{3}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 2.5

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

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Step 2.5.1

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

$x - 3 = 0$

Step 2.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.6

The final solution is all the values that make $x (3 x - 1) (x - 3) = 0$ true.

$x = 0, \frac{1}{3}, 3$

Step 3

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