Finite Math Examples

$f (x) = x^{4} + 10 x^{3} - 12 x^{2} - 10 x + 11$

Step 1

Set $x^{4} + 10 x^{3} - 12 x^{2} - 10 x + 11$ equal to $0$ .

$x^{4} + 10 x^{3} - 12 x^{2} - 10 x + 11 = 0$

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

Regroup terms.

$10 x^{3} - 10 x + x^{4} - 12 x^{2} + 11 = 0$

Step 2.1.2

Factor $10 x$ out of $10 x^{3} - 10 x$ .

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Step 2.1.2.1

Factor $10 x$ out of $10 x^{3}$ .

$10 x (x^{2}) - 10 x + x^{4} - 12 x^{2} + 11 = 0$

Step 2.1.2.2

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

$10 x (x^{2}) + 10 x (- 1) + x^{4} - 12 x^{2} + 11 = 0$

Step 2.1.2.3

Factor $10 x$ out of $10 x (x^{2}) + 10 x (- 1)$ .

$10 x (x^{2} - 1) + x^{4} - 12 x^{2} + 11 = 0$

Step 2.1.3

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

$10 x (x^{2} - 1^{2}) + x^{4} - 12 x^{2} + 11 = 0$

Step 2.1.4

Factor.

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Step 2.1.4.1

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

$10 x ((x + 1) (x - 1)) + x^{4} - 12 x^{2} + 11 = 0$

Step 2.1.4.2

Remove unnecessary parentheses.

$10 x (x + 1) (x - 1) + x^{4} - 12 x^{2} + 11 = 0$

Step 2.1.5

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$10 x (x + 1) (x - 1) + {(x^{2})}^{2} - 12 x^{2} + 11 = 0$

Step 2.1.6

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$10 x (x + 1) (x - 1) + u^{2} - 12 u + 11 = 0$

Step 2.1.7

Factor $u^{2} - 12 u + 11$ using the AC method.

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Step 2.1.7.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $11$ and whose sum is $- 12$ .

$- 11, - 1$

Step 2.1.7.2

Write the factored form using these integers.

$10 x (x + 1) (x - 1) + (u - 11) (u - 1) = 0$

Step 2.1.8

Replace all occurrences of $u$ with $x^{2}$ .

$10 x (x + 1) (x - 1) + (x^{2} - 11) (x^{2} - 1) = 0$

Step 2.1.9

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

$10 x (x + 1) (x - 1) + (x^{2} - 11) (x^{2} - 1^{2}) = 0$

Step 2.1.10

Factor.

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Step 2.1.10.1

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

$10 x (x + 1) (x - 1) + (x^{2} - 11) ((x + 1) (x - 1)) = 0$

Step 2.1.10.2

Remove unnecessary parentheses.

$10 x (x + 1) (x - 1) + (x^{2} - 11) (x + 1) (x - 1) = 0$

Step 2.1.11

Factor $(x + 1) (x - 1)$ out of $10 x (x + 1) (x - 1) + (x^{2} - 11) (x + 1) (x - 1)$ .

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Step 2.1.11.1

Factor $(x + 1) (x - 1)$ out of $10 x (x + 1) (x - 1)$ .

$(x + 1) (x - 1) (10 x) + (x^{2} - 11) (x + 1) (x - 1) = 0$

Step 2.1.11.2

Factor $(x + 1) (x - 1)$ out of $(x^{2} - 11) (x + 1) (x - 1)$ .

$(x + 1) (x - 1) (10 x) + (x + 1) (x - 1) (x^{2} - 11) = 0$

Step 2.1.11.3

Factor $(x + 1) (x - 1)$ out of $(x + 1) (x - 1) (10 x) + (x + 1) (x - 1) (x^{2} - 11)$ .

$(x + 1) (x - 1) (10 x + x^{2} - 11) = 0$

Step 2.1.12

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(x + 1) (x - 1) (10 u + u^{2} - 11) = 0$

Step 2.1.13

Factor $10 u + u^{2} - 11$ using the AC method.

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Step 2.1.13.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 11$ and whose sum is $10$ .

$- 1, 11$

Step 2.1.13.2

Write the factored form using these integers.

$(x + 1) (x - 1) ((u - 1) (u + 11)) = 0$

Step 2.1.14

Factor.

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Step 2.1.14.1

Replace all occurrences of $u$ with $x$ .

$(x + 1) (x - 1) ((x - 1) (x + 11)) = 0$

Step 2.1.14.2

Remove unnecessary parentheses.

$(x + 1) (x - 1) (x - 1) (x + 11) = 0$

Step 2.1.15

Combine exponents.

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Step 2.1.15.1

Raise $x - 1$ to the power of $1$ .

$(x + 1) ((x - 1) (x - 1)) (x + 11) = 0$

Step 2.1.15.2

Raise $x - 1$ to the power of $1$ .

$(x + 1) ((x - 1) (x - 1)) (x + 11) = 0$

Step 2.1.15.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x + 1) {(x - 1)}^{1 + 1} (x + 11) = 0$

Step 2.1.15.4

Add $1$ and $1$ .

$(x + 1) {(x - 1)}^{2} (x + 11) = 0$

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 $0$ .

$x + 1 = 0$

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

$x + 11 = 0$

Step 2.3

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

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Step 2.3.1

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

$x + 1 = 0$

Step 2.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.4

Set ${(x - 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set ${(x - 1)}^{2}$ equal to $0$ .

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

Step 2.4.2

Solve ${(x - 1)}^{2} = 0$ for $x$ .

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Step 2.4.2.1

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

$x - 1 = 0$

Step 2.4.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.5

Set $x + 11$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

Step 2.5.2

Subtract $11$ from both sides of the equation.

$x = - 11$

Step 2.6

The final solution is all the values that make $(x + 1) {(x - 1)}^{2} (x + 11) = 0$ true.

$x = - 1, 1, - 11$

Step 3