Precalculus Examples

$n^{3} - 3 n^{2} - n + 3 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

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

Step 1.1.2

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

$n^{2} (n - 3) - (n - 3) = 0$

Step 1.2

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

$(n - 3) (n^{2} - 1) = 0$

Step 1.3

Rewrite $1$ as $1^{2}$ .

$(n - 3) (n^{2} - 1^{2}) = 0$

Step 1.4

Factor.

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Step 1.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = n$ and $b = 1$ .

$(n - 3) ((n + 1) (n - 1)) = 0$

Step 1.4.2

Remove unnecessary parentheses.

$(n - 3) (n + 1) (n - 1) = 0$

Step 2

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

$n - 3 = 0$

$n + 1 = 0$

$n - 1 = 0$

Step 3

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

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Step 3.1

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

$n - 3 = 0$

Step 3.2

Add $3$ to both sides of the equation.

$n = 3$

Step 4

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

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Step 4.1

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

$n + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$n = - 1$

Step 5

Set $n - 1$ equal to $0$ and solve for $n$ .

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Step 5.1

Set $n - 1$ equal to $0$ .

$n - 1 = 0$

Step 5.2

Add $1$ to both sides of the equation.

$n = 1$

Step 6

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

$n = 3, - 1, 1$