Algebra Examples

$\frac{6 x^{3} - 10 x^{2}}{3 x^{3} - 2 x^{2} - 5 x}$

Step 1

Set the denominator in $\frac{6 x^{3} - 10 x^{2}}{3 x^{3} - 2 x^{2} - 5 x}$ equal to $0$ to find where the expression is undefined.

$3 x^{3} - 2 x^{2} - 5 x = 0$

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 $3 x^{3} - 2 x^{2} - 5 x$ .

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Step 2.1.1.1

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

$x (3 x^{2}) - 2 x^{2} - 5 x = 0$

Step 2.1.1.2

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

$x (3 x^{2}) + x (- 2 x) - 5 x = 0$

Step 2.1.1.3

Factor $x$ out of $- 5 x$ .

$x (3 x^{2}) + x (- 2 x) + x \cdot - 5 = 0$

Step 2.1.1.4

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

$x (3 x^{2} - 2 x) + x \cdot - 5 = 0$

Step 2.1.1.5

Factor $x$ out of $x (3 x^{2} - 2 x) + x \cdot - 5$ .

$x (3 x^{2} - 2 x - 5) = 0$

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 $a \cdot c = 3 \cdot - 5 = - 15$ and whose sum is $b = - 2$ .

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Step 2.1.2.1.1.1

Factor $- 2$ out of $- 2 x$ .

$x (3 x^{2} - 2 x - 5) = 0$

Step 2.1.2.1.1.2

Rewrite $- 2$ as $3$ plus $- 5$

$x (3 x^{2} + (3 - 5) x - 5) = 0$

Step 2.1.2.1.1.3

Apply the distributive property.

$x (3 x^{2} + 3 x - 5 x - 5) = 0$

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.

$x ((3 x^{2} + 3 x) - 5 x - 5) = 0$

Step 2.1.2.1.2.2

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

$x (3 x (x + 1) - 5 (x + 1)) = 0$

Step 2.1.2.1.3

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

$x ((x + 1) (3 x - 5)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$x (x + 1) (3 x - 5) = 0$

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 $0$ .

$x = 0$

$x + 1 = 0$

$3 x - 5 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.5

Set $3 x - 5$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $3 x - 5$ equal to $0$ .

$3 x - 5 = 0$

Step 2.5.2

Solve $3 x - 5 = 0$ for $x$ .

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Step 2.5.2.1

Add $5$ to both sides of the equation.

$3 x = 5$

Step 2.5.2.2

Divide each term in $3 x = 5$ by $3$ and simplify.

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Step 2.5.2.2.1

Divide each term in $3 x = 5$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3}$

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{3}$

Step 2.6

The final solution is all the values that make $x (x + 1) (3 x - 5) = 0$ true.

$x = 0, - 1, \frac{5}{3}$

Step 3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 1, x = 0, x = \frac{5}{3}$

Step 4