Finite Math Examples

$(12 x - 1) \cdot (6 x - 1) \cdot (4 x - 1) \cdot (3 x - 1) = 5$

Step 1

Simplify $(12 x - 1) \cdot (6 x - 1) \cdot (4 x - 1) \cdot (3 x - 1)$ .

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Step 1.1

Expand $(12 x - 1) (6 x - 1)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$(12 x (6 x - 1) - 1 (6 x - 1)) \cdot (4 x - 1) \cdot (3 x - 1) = 5$

Step 1.1.2

Apply the distributive property.

$(12 x (6 x) + 12 x \cdot - 1 - 1 (6 x - 1)) \cdot (4 x - 1) \cdot (3 x - 1) = 5$

Step 1.1.3

Apply the distributive property.

$(12 x (6 x) + 12 x \cdot - 1 - 1 (6 x) - 1 \cdot - 1) \cdot (4 x - 1) \cdot (3 x - 1) = 5$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite using the commutative property of multiplication.

$(12 \cdot 6 x \cdot x + 12 x \cdot - 1 - 1 (6 x) - 1 \cdot - 1) \cdot (4 x - 1) \cdot (3 x - 1) = 5$

Step 1.2.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.1.2.1

Move $x$ .

$(12 \cdot 6 (x \cdot x) + 12 x \cdot - 1 - 1 (6 x) - 1 \cdot - 1) \cdot (4 x - 1) \cdot (3 x - 1) = 5$

Step 1.2.1.2.2

Multiply $x$ by $x$ .

$(12 \cdot 6 x^{2} + 12 x \cdot - 1 - 1 (6 x) - 1 \cdot - 1) \cdot (4 x - 1) \cdot (3 x - 1) = 5$

Step 1.2.1.3

Multiply $12$ by $6$ .

$(72 x^{2} + 12 x \cdot - 1 - 1 (6 x) - 1 \cdot - 1) \cdot (4 x - 1) \cdot (3 x - 1) = 5$

Step 1.2.1.4

Multiply $- 1$ by $12$ .

$(72 x^{2} - 12 x - 1 (6 x) - 1 \cdot - 1) \cdot (4 x - 1) \cdot (3 x - 1) = 5$

Step 1.2.1.5

Multiply $6$ by $- 1$ .

$(72 x^{2} - 12 x - 6 x - 1 \cdot - 1) \cdot (4 x - 1) \cdot (3 x - 1) = 5$

Step 1.2.1.6

Multiply $- 1$ by $- 1$ .

$(72 x^{2} - 12 x - 6 x + 1) \cdot (4 x - 1) \cdot (3 x - 1) = 5$

Step 1.2.2

Subtract $6 x$ from $- 12 x$ .

$(72 x^{2} - 18 x + 1) \cdot (4 x - 1) \cdot (3 x - 1) = 5$

Step 1.3

Expand $(72 x^{2} - 18 x + 1) (4 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$(72 x^{2} (4 x) + 72 x^{2} \cdot - 1 - 18 x (4 x) - 18 x \cdot - 1 + 1 (4 x) + 1 \cdot - 1) \cdot (3 x - 1) = 5$

Step 1.4

Simplify terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Rewrite using the commutative property of multiplication.

$(72 \cdot 4 x^{2} x + 72 x^{2} \cdot - 1 - 18 x (4 x) - 18 x \cdot - 1 + 1 (4 x) + 1 \cdot - 1) \cdot (3 x - 1) = 5$

Step 1.4.1.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.4.1.2.1

Move $x$ .

$(72 \cdot 4 (x \cdot x^{2}) + 72 x^{2} \cdot - 1 - 18 x (4 x) - 18 x \cdot - 1 + 1 (4 x) + 1 \cdot - 1) \cdot (3 x - 1) = 5$

Step 1.4.1.2.2

Multiply $x$ by $x^{2}$ .

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Step 1.4.1.2.2.1

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

$(72 \cdot 4 (x^{1} x^{2}) + 72 x^{2} \cdot - 1 - 18 x (4 x) - 18 x \cdot - 1 + 1 (4 x) + 1 \cdot - 1) \cdot (3 x - 1) = 5$

Step 1.4.1.2.2.2

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

$(72 \cdot 4 x^{1 + 2} + 72 x^{2} \cdot - 1 - 18 x (4 x) - 18 x \cdot - 1 + 1 (4 x) + 1 \cdot - 1) \cdot (3 x - 1) = 5$

Step 1.4.1.2.3

Add $1$ and $2$ .

$(72 \cdot 4 x^{3} + 72 x^{2} \cdot - 1 - 18 x (4 x) - 18 x \cdot - 1 + 1 (4 x) + 1 \cdot - 1) \cdot (3 x - 1) = 5$

Step 1.4.1.3

Multiply $72$ by $4$ .

$(288 x^{3} + 72 x^{2} \cdot - 1 - 18 x (4 x) - 18 x \cdot - 1 + 1 (4 x) + 1 \cdot - 1) \cdot (3 x - 1) = 5$

Step 1.4.1.4

Multiply $- 1$ by $72$ .

$(288 x^{3} - 72 x^{2} - 18 x (4 x) - 18 x \cdot - 1 + 1 (4 x) + 1 \cdot - 1) \cdot (3 x - 1) = 5$

Step 1.4.1.5

Rewrite using the commutative property of multiplication.

$(288 x^{3} - 72 x^{2} - 18 \cdot 4 x \cdot x - 18 x \cdot - 1 + 1 (4 x) + 1 \cdot - 1) \cdot (3 x - 1) = 5$

Step 1.4.1.6

Multiply $x$ by $x$ by adding the exponents.

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Step 1.4.1.6.1

Move $x$ .

$(288 x^{3} - 72 x^{2} - 18 \cdot 4 (x \cdot x) - 18 x \cdot - 1 + 1 (4 x) + 1 \cdot - 1) \cdot (3 x - 1) = 5$

Step 1.4.1.6.2

Multiply $x$ by $x$ .

$(288 x^{3} - 72 x^{2} - 18 \cdot 4 x^{2} - 18 x \cdot - 1 + 1 (4 x) + 1 \cdot - 1) \cdot (3 x - 1) = 5$

Step 1.4.1.7

Multiply $- 18$ by $4$ .

$(288 x^{3} - 72 x^{2} - 72 x^{2} - 18 x \cdot - 1 + 1 (4 x) + 1 \cdot - 1) \cdot (3 x - 1) = 5$

Step 1.4.1.8

Multiply $- 1$ by $- 18$ .

$(288 x^{3} - 72 x^{2} - 72 x^{2} + 18 x + 1 (4 x) + 1 \cdot - 1) \cdot (3 x - 1) = 5$

Step 1.4.1.9

Multiply $4 x$ by $1$ .

$(288 x^{3} - 72 x^{2} - 72 x^{2} + 18 x + 4 x + 1 \cdot - 1) \cdot (3 x - 1) = 5$

Step 1.4.1.10

Multiply $- 1$ by $1$ .

$(288 x^{3} - 72 x^{2} - 72 x^{2} + 18 x + 4 x - 1) \cdot (3 x - 1) = 5$

Step 1.4.2

Simplify by adding terms.

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Step 1.4.2.1

Subtract $72 x^{2}$ from $- 72 x^{2}$ .

$(288 x^{3} - 144 x^{2} + 18 x + 4 x - 1) \cdot (3 x - 1) = 5$

Step 1.4.2.2

Add $18 x$ and $4 x$ .

$(288 x^{3} - 144 x^{2} + 22 x - 1) \cdot (3 x - 1) = 5$

Step 1.5

Expand $(288 x^{3} - 144 x^{2} + 22 x - 1) (3 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$288 x^{3} (3 x) + 288 x^{3} \cdot - 1 - 144 x^{2} (3 x) - 144 x^{2} \cdot - 1 + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6

Simplify terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Rewrite using the commutative property of multiplication.

$288 \cdot 3 x^{3} x + 288 x^{3} \cdot - 1 - 144 x^{2} (3 x) - 144 x^{2} \cdot - 1 + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.2

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 1.6.1.2.1

Move $x$ .

$288 \cdot 3 (x \cdot x^{3}) + 288 x^{3} \cdot - 1 - 144 x^{2} (3 x) - 144 x^{2} \cdot - 1 + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.2.2

Multiply $x$ by $x^{3}$ .

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Step 1.6.1.2.2.1

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

$288 \cdot 3 (x^{1} x^{3}) + 288 x^{3} \cdot - 1 - 144 x^{2} (3 x) - 144 x^{2} \cdot - 1 + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.2.2.2

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

$288 \cdot 3 x^{1 + 3} + 288 x^{3} \cdot - 1 - 144 x^{2} (3 x) - 144 x^{2} \cdot - 1 + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.2.3

Add $1$ and $3$ .

$288 \cdot 3 x^{4} + 288 x^{3} \cdot - 1 - 144 x^{2} (3 x) - 144 x^{2} \cdot - 1 + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.3

Multiply $288$ by $3$ .

$864 x^{4} + 288 x^{3} \cdot - 1 - 144 x^{2} (3 x) - 144 x^{2} \cdot - 1 + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.4

Multiply $- 1$ by $288$ .

$864 x^{4} - 288 x^{3} - 144 x^{2} (3 x) - 144 x^{2} \cdot - 1 + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.5

Rewrite using the commutative property of multiplication.

$864 x^{4} - 288 x^{3} - 144 \cdot 3 x^{2} x - 144 x^{2} \cdot - 1 + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.6

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.6.1.6.1

Move $x$ .

$864 x^{4} - 288 x^{3} - 144 \cdot 3 (x \cdot x^{2}) - 144 x^{2} \cdot - 1 + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.6.2

Multiply $x$ by $x^{2}$ .

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Step 1.6.1.6.2.1

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

$864 x^{4} - 288 x^{3} - 144 \cdot 3 (x^{1} x^{2}) - 144 x^{2} \cdot - 1 + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.6.2.2

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

$864 x^{4} - 288 x^{3} - 144 \cdot 3 x^{1 + 2} - 144 x^{2} \cdot - 1 + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.6.3

Add $1$ and $2$ .

$864 x^{4} - 288 x^{3} - 144 \cdot 3 x^{3} - 144 x^{2} \cdot - 1 + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.7

Multiply $- 144$ by $3$ .

$864 x^{4} - 288 x^{3} - 432 x^{3} - 144 x^{2} \cdot - 1 + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.8

Multiply $- 1$ by $- 144$ .

$864 x^{4} - 288 x^{3} - 432 x^{3} + 144 x^{2} + 22 x (3 x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.9

Rewrite using the commutative property of multiplication.

$864 x^{4} - 288 x^{3} - 432 x^{3} + 144 x^{2} + 22 \cdot 3 x \cdot x + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.10

Multiply $x$ by $x$ by adding the exponents.

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Step 1.6.1.10.1

Move $x$ .

$864 x^{4} - 288 x^{3} - 432 x^{3} + 144 x^{2} + 22 \cdot 3 (x \cdot x) + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.10.2

Multiply $x$ by $x$ .

$864 x^{4} - 288 x^{3} - 432 x^{3} + 144 x^{2} + 22 \cdot 3 x^{2} + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.11

Multiply $22$ by $3$ .

$864 x^{4} - 288 x^{3} - 432 x^{3} + 144 x^{2} + 66 x^{2} + 22 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.12

Multiply $- 1$ by $22$ .

$864 x^{4} - 288 x^{3} - 432 x^{3} + 144 x^{2} + 66 x^{2} - 22 x - 1 (3 x) - 1 \cdot - 1 = 5$

Step 1.6.1.13

Multiply $3$ by $- 1$ .

$864 x^{4} - 288 x^{3} - 432 x^{3} + 144 x^{2} + 66 x^{2} - 22 x - 3 x - 1 \cdot - 1 = 5$

Step 1.6.1.14

Multiply $- 1$ by $- 1$ .

$864 x^{4} - 288 x^{3} - 432 x^{3} + 144 x^{2} + 66 x^{2} - 22 x - 3 x + 1 = 5$

Step 1.6.2

Simplify by adding terms.

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Step 1.6.2.1

Subtract $432 x^{3}$ from $- 288 x^{3}$ .

$864 x^{4} - 720 x^{3} + 144 x^{2} + 66 x^{2} - 22 x - 3 x + 1 = 5$

Step 1.6.2.2

Add $144 x^{2}$ and $66 x^{2}$ .

$864 x^{4} - 720 x^{3} + 210 x^{2} - 22 x - 3 x + 1 = 5$

Step 1.6.2.3

Subtract $3 x$ from $- 22 x$ .

$864 x^{4} - 720 x^{3} + 210 x^{2} - 25 x + 1 = 5$

Step 2

Subtract $5$ from both sides of the equation.

$864 x^{4} - 720 x^{3} + 210 x^{2} - 25 x + 1 - 5 = 0$

Step 3

Subtract $5$ from $1$ .

$864 x^{4} - 720 x^{3} + 210 x^{2} - 25 x - 4 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $864 x^{4} - 720 x^{3} + 210 x^{2} - 25 x - 4$ using the rational roots test.

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Step 4.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 4, \pm 2$

$q = \pm 1, \pm 864, \pm 2, \pm 432, \pm 3, \pm 288, \pm 4, \pm 216, \pm 6, \pm 144, \pm 8, \pm 108, \pm 9, \pm 96, \pm 12, \pm 72, \pm 16, \pm 54, \pm 18, \pm 48, \pm 24, \pm 36, \pm 27, \pm 32$

Step 4.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.0011574, \pm 0.5, \pm 0.0023\overline{148}, \pm 0.\overline{3}, \pm 0.00347\overline{2}, \pm 0.25, \pm 0.004\overline{629}, \pm 0.1\overline{6}, \pm 0.0069\overline{4}, \pm 0.125, \pm 0.00\overline{925}, \pm 0.\overline{1}, \pm 0.01041\overline{6}, \pm 0.08\overline{3}, \pm 0.013\overline{8}, \pm 0.0625, \pm 0.0\overline{185}, \pm 0.0\overline{5}, \pm 0.0208\overline{3}, \pm 0.041\overline{6}, \pm 0.02\overline{7}, \pm 0.\overline{037}, \pm 0.03125, \pm 4, \pm 2, \pm 1.\overline{3}, \pm 0.\overline{6}, \pm 0.\overline{4}, \pm 0.\overline{074}, \pm 0.\overline{2}, \pm 0.\overline{148}$

Step 4.1.3

Substitute $- 0.08\overline{3}$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.08\overline{3}$ is a root of the polynomial.

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Step 4.1.3.1

Substitute $- 0.08\overline{3}$ into the polynomial.

$864 {(- 0.08\overline{3})}^{4} - 720 {(- 0.08\overline{3})}^{3} + 210 {(- 0.08\overline{3})}^{2} - 25 \cdot - 0.08\overline{3} - 4$

Step 4.1.3.2

Raise $- 0.08\overline{3}$ to the power of $4$ .

$864 \cdot 0.00004822 - 720 {(- 0.08\overline{3})}^{3} + 210 {(- 0.08\overline{3})}^{2} - 25 \cdot - 0.08\overline{3} - 4$

Step 4.1.3.3

Multiply $864$ by $0.00004822$ .

$0.04166666 - 720 {(- 0.08\overline{3})}^{3} + 210 {(- 0.08\overline{3})}^{2} - 25 \cdot - 0.08\overline{3} - 4$

Step 4.1.3.4

Raise $- 0.08\overline{3}$ to the power of $3$ .

$0.04166666 - 720 \cdot - 0.0005787 + 210 {(- 0.08\overline{3})}^{2} - 25 \cdot - 0.08\overline{3} - 4$

Step 4.1.3.5

Multiply $- 720$ by $- 0.0005787$ .

$0.04166666 + 0.41666666 + 210 {(- 0.08\overline{3})}^{2} - 25 \cdot - 0.08\overline{3} - 4$

Step 4.1.3.6

Add $0.04166666$ and $0.41666666$ .

$0.45833333 + 210 {(- 0.08\overline{3})}^{2} - 25 \cdot - 0.08\overline{3} - 4$

Step 4.1.3.7

Raise $- 0.08\overline{3}$ to the power of $2$ .

$0.45833333 + 210 \cdot 0.0069\overline{4} - 25 \cdot - 0.08\overline{3} - 4$

Step 4.1.3.8

Multiply $210$ by $0.0069\overline{4}$ .

$0.45833333 + 1.45833333 - 25 \cdot - 0.08\overline{3} - 4$

Step 4.1.3.9

Add $0.45833333$ and $1.45833333$ .

$1.91666666 - 25 \cdot - 0.08\overline{3} - 4$

Step 4.1.3.10

Multiply $- 25$ by $- 0.08\overline{3}$ .

$1.91666666 + 2.08333333 - 4$

Step 4.1.3.11

Add $1.91666666$ and $2.08333333$ .

$4 - 4$

Step 4.1.3.12

Subtract $4$ from $4$ .

$0$

Step 4.1.4

Since $- 0.08\overline{3}$ is a known root, divide the polynomial by $12 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{864 x^{4} - 720 x^{3} + 210 x^{2} - 25 x - 4}{12 x + 1}$

Step 4.1.5

Divide $864 x^{4} - 720 x^{3} + 210 x^{2} - 25 x - 4$ by $12 x + 1$ .

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Step 4.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

Step 4.1.5.2

Divide the highest order term in the dividend $864 x^{4}$ by the highest order term in divisor $12 x$ .

$72 x^{3}$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

Step 4.1.5.3

Multiply the new quotient term by the divisor.

$72 x^{3}$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

$864 x^{4}$

$72 x^{3}$

Step 4.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $864 x^{4} + 72 x^{3}$

				$72 x^{3}$
$12 x$	+	$1$		$864 x^{4}$	-	$720 x^{3}$	+	$210 x^{2}$	-	$25 x$	-	$4$
			-	$864 x^{4}$	-	$72 x^{3}$

Step 4.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$72 x^{3}$
$12 x$	+	$1$		$864 x^{4}$	-	$720 x^{3}$	+	$210 x^{2}$	-	$25 x$	-	$4$
			-	$864 x^{4}$	-	$72 x^{3}$
					-	$792 x^{3}$

Step 4.1.5.6

Pull the next terms from the original dividend down into the current dividend.

$72 x^{3}$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

$864 x^{4}$

$72 x^{3}$

$792 x^{3}$

$210 x^{2}$

Step 4.1.5.7

Divide the highest order term in the dividend $- 792 x^{3}$ by the highest order term in divisor $12 x$ .

$72 x^{3}$

$66 x^{2}$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

$864 x^{4}$

$72 x^{3}$

$792 x^{3}$

$210 x^{2}$

Step 4.1.5.8

Multiply the new quotient term by the divisor.

				$72 x^{3}$	-	$66 x^{2}$
$12 x$	+	$1$		$864 x^{4}$	-	$720 x^{3}$	+	$210 x^{2}$	-	$25 x$	-	$4$
			-	$864 x^{4}$	-	$72 x^{3}$
					-	$792 x^{3}$	+	$210 x^{2}$
					-	$792 x^{3}$	-	$66 x^{2}$

Step 4.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 792 x^{3} - 66 x^{2}$

$72 x^{3}$

$66 x^{2}$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

$864 x^{4}$

$72 x^{3}$

$792 x^{3}$

$210 x^{2}$

$792 x^{3}$

$66 x^{2}$

Step 4.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$72 x^{3}$

$66 x^{2}$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

$864 x^{4}$

$72 x^{3}$

$792 x^{3}$

$210 x^{2}$

$792 x^{3}$

$66 x^{2}$

$276 x^{2}$

Step 4.1.5.11

Pull the next terms from the original dividend down into the current dividend.

$72 x^{3}$

$66 x^{2}$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

$864 x^{4}$

$72 x^{3}$

$792 x^{3}$

$210 x^{2}$

$792 x^{3}$

$66 x^{2}$

$276 x^{2}$

$25 x$

Step 4.1.5.12

Divide the highest order term in the dividend $276 x^{2}$ by the highest order term in divisor $12 x$ .

$72 x^{3}$

$66 x^{2}$

$23 x$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

$864 x^{4}$

$72 x^{3}$

$792 x^{3}$

$210 x^{2}$

$792 x^{3}$

$66 x^{2}$

$276 x^{2}$

$25 x$

Step 4.1.5.13

Multiply the new quotient term by the divisor.

$72 x^{3}$

$66 x^{2}$

$23 x$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

$864 x^{4}$

$72 x^{3}$

$792 x^{3}$

$210 x^{2}$

$792 x^{3}$

$66 x^{2}$

$276 x^{2}$

$25 x$

$276 x^{2}$

$23 x$

Step 4.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $276 x^{2} + 23 x$

$72 x^{3}$

$66 x^{2}$

$23 x$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

$864 x^{4}$

$72 x^{3}$

$792 x^{3}$

$210 x^{2}$

$792 x^{3}$

$66 x^{2}$

$276 x^{2}$

$25 x$

$276 x^{2}$

$23 x$

Step 4.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$72 x^{3}$

$66 x^{2}$

$23 x$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

$864 x^{4}$

$72 x^{3}$

$792 x^{3}$

$210 x^{2}$

$792 x^{3}$

$66 x^{2}$

$276 x^{2}$

$25 x$

$276 x^{2}$

$23 x$

$48 x$

Step 4.1.5.16

Pull the next terms from the original dividend down into the current dividend.

$72 x^{3}$

$66 x^{2}$

$23 x$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

$864 x^{4}$

$72 x^{3}$

$792 x^{3}$

$210 x^{2}$

$792 x^{3}$

$66 x^{2}$

$276 x^{2}$

$25 x$

$276 x^{2}$

$23 x$

$48 x$

$4$

Step 4.1.5.17

Divide the highest order term in the dividend $- 48 x$ by the highest order term in divisor $12 x$ .

$72 x^{3}$

$66 x^{2}$

$23 x$

$4$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

$864 x^{4}$

$72 x^{3}$

$792 x^{3}$

$210 x^{2}$

$792 x^{3}$

$66 x^{2}$

$276 x^{2}$

$25 x$

$276 x^{2}$

$23 x$

$48 x$

$4$

Step 4.1.5.18

Multiply the new quotient term by the divisor.

				$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$
$12 x$	+	$1$		$864 x^{4}$	-	$720 x^{3}$	+	$210 x^{2}$	-	$25 x$	-	$4$
			-	$864 x^{4}$	-	$72 x^{3}$
					-	$792 x^{3}$	+	$210 x^{2}$
					+	$792 x^{3}$	+	$66 x^{2}$
							+	$276 x^{2}$	-	$25 x$
							-	$276 x^{2}$	-	$23 x$
									-	$48 x$	-	$4$
									-	$48 x$	-	$4$

Step 4.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 48 x - 4$

$72 x^{3}$

$66 x^{2}$

$23 x$

$4$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

$864 x^{4}$

$72 x^{3}$

$792 x^{3}$

$210 x^{2}$

$792 x^{3}$

$66 x^{2}$

$276 x^{2}$

$25 x$

$276 x^{2}$

$23 x$

$48 x$

$4$

$48 x$

$4$

Step 4.1.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$72 x^{3}$

$66 x^{2}$

$23 x$

$4$

$12 x$

$1$

$864 x^{4}$

$720 x^{3}$

$210 x^{2}$

$25 x$

$4$

$864 x^{4}$

$72 x^{3}$

$792 x^{3}$

$210 x^{2}$

$792 x^{3}$

$66 x^{2}$

$276 x^{2}$

$25 x$

$276 x^{2}$

$23 x$

$48 x$

$4$

$48 x$

$4$

$0$

Step 4.1.5.21

Since the remander is $0$ , the final answer is the quotient.

$72 x^{3} - 66 x^{2} + 23 x - 4$

Step 4.1.6

Write $864 x^{4} - 720 x^{3} + 210 x^{2} - 25 x - 4$ as a set of factors.

$(12 x + 1) (72 x^{3} - 66 x^{2} + 23 x - 4) = 0$

Step 4.2

Factor $72 x^{3} - 66 x^{2} + 23 x - 4$ using the rational roots test.

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Step 4.2.1

Factor $72 x^{3} - 66 x^{2} + 23 x - 4$ using the rational roots test.

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Step 4.2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 4, \pm 2$

$q = \pm 1, \pm 72, \pm 2, \pm 36, \pm 3, \pm 24, \pm 4, \pm 18, \pm 6, \pm 12, \pm 8, \pm 9$

Step 4.2.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.013\overline{8}, \pm 0.5, \pm 0.02\overline{7}, \pm 0.\overline{3}, \pm 0.041\overline{6}, \pm 0.25, \pm 0.0\overline{5}, \pm 0.1\overline{6}, \pm 0.08\overline{3}, \pm 0.125, \pm 0.\overline{1}, \pm 4, \pm 2, \pm 1.\overline{3}, \pm 0.\overline{2}, \pm 0.\overline{6}, \pm 0.\overline{4}$

Step 4.2.1.3

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

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Step 4.2.1.3.1

Substitute $0.5$ into the polynomial.

$72 \cdot {0.5}^{3} - 66 \cdot {0.5}^{2} + 23 \cdot 0.5 - 4$

Step 4.2.1.3.2

Raise $0.5$ to the power of $3$ .

$72 \cdot 0.125 - 66 \cdot {0.5}^{2} + 23 \cdot 0.5 - 4$

Step 4.2.1.3.3

Multiply $72$ by $0.125$ .

$9 - 66 \cdot {0.5}^{2} + 23 \cdot 0.5 - 4$

Step 4.2.1.3.4

Raise $0.5$ to the power of $2$ .

$9 - 66 \cdot 0.25 + 23 \cdot 0.5 - 4$

Step 4.2.1.3.5

Multiply $- 66$ by $0.25$ .

$9 - 16.5 + 23 \cdot 0.5 - 4$

Step 4.2.1.3.6

Subtract $16.5$ from $9$ .

$- 7.5 + 23 \cdot 0.5 - 4$

Step 4.2.1.3.7

Multiply $23$ by $0.5$ .

$- 7.5 + 11.5 - 4$

Step 4.2.1.3.8

Add $- 7.5$ and $11.5$ .

$4 - 4$

Step 4.2.1.3.9

Subtract $4$ from $4$ .

$0$

Step 4.2.1.4

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{72 x^{3} - 66 x^{2} + 23 x - 4}{2 x - 1}$

Step 4.2.1.5

Divide $72 x^{3} - 66 x^{2} + 23 x - 4$ by $2 x - 1$ .

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Step 4.2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$2 x$

$1$

$72 x^{3}$

$66 x^{2}$

$23 x$

$4$

Step 4.2.1.5.2

Divide the highest order term in the dividend $72 x^{3}$ by the highest order term in divisor $2 x$ .

					$36 x^{2}$
	$2 x$	-	$1$		$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$

Step 4.2.1.5.3

Multiply the new quotient term by the divisor.

				$36 x^{2}$
$2 x$	-	$1$		$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$
			+	$72 x^{3}$	-	$36 x^{2}$

Step 4.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $72 x^{3} - 36 x^{2}$

				$36 x^{2}$
$2 x$	-	$1$		$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$
			-	$72 x^{3}$	+	$36 x^{2}$

Step 4.2.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$36 x^{2}$
$2 x$	-	$1$		$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$
			-	$72 x^{3}$	+	$36 x^{2}$
					-	$30 x^{2}$

Step 4.2.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$36 x^{2}$
$2 x$	-	$1$		$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$
			-	$72 x^{3}$	+	$36 x^{2}$
					-	$30 x^{2}$	+	$23 x$

Step 4.2.1.5.7

Divide the highest order term in the dividend $- 30 x^{2}$ by the highest order term in divisor $2 x$ .

				$36 x^{2}$	-	$15 x$
$2 x$	-	$1$		$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$
			-	$72 x^{3}$	+	$36 x^{2}$
					-	$30 x^{2}$	+	$23 x$

Step 4.2.1.5.8

Multiply the new quotient term by the divisor.

				$36 x^{2}$	-	$15 x$
$2 x$	-	$1$		$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$
			-	$72 x^{3}$	+	$36 x^{2}$
					-	$30 x^{2}$	+	$23 x$
					-	$30 x^{2}$	+	$15 x$

Step 4.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 30 x^{2} + 15 x$

				$36 x^{2}$	-	$15 x$
$2 x$	-	$1$		$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$
			-	$72 x^{3}$	+	$36 x^{2}$
					-	$30 x^{2}$	+	$23 x$
					+	$30 x^{2}$	-	$15 x$

Step 4.2.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$36 x^{2}$	-	$15 x$
$2 x$	-	$1$		$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$
			-	$72 x^{3}$	+	$36 x^{2}$
					-	$30 x^{2}$	+	$23 x$
					+	$30 x^{2}$	-	$15 x$
							+	$8 x$

Step 4.2.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$36 x^{2}$	-	$15 x$
$2 x$	-	$1$		$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$
			-	$72 x^{3}$	+	$36 x^{2}$
					-	$30 x^{2}$	+	$23 x$
					+	$30 x^{2}$	-	$15 x$
							+	$8 x$	-	$4$

Step 4.2.1.5.12

Divide the highest order term in the dividend $8 x$ by the highest order term in divisor $2 x$ .

				$36 x^{2}$	-	$15 x$	+	$4$
$2 x$	-	$1$		$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$
			-	$72 x^{3}$	+	$36 x^{2}$
					-	$30 x^{2}$	+	$23 x$
					+	$30 x^{2}$	-	$15 x$
							+	$8 x$	-	$4$

Step 4.2.1.5.13

Multiply the new quotient term by the divisor.

				$36 x^{2}$	-	$15 x$	+	$4$
$2 x$	-	$1$		$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$
			-	$72 x^{3}$	+	$36 x^{2}$
					-	$30 x^{2}$	+	$23 x$
					+	$30 x^{2}$	-	$15 x$
							+	$8 x$	-	$4$
							+	$8 x$	-	$4$

Step 4.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $8 x - 4$

				$36 x^{2}$	-	$15 x$	+	$4$
$2 x$	-	$1$		$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$
			-	$72 x^{3}$	+	$36 x^{2}$
					-	$30 x^{2}$	+	$23 x$
					+	$30 x^{2}$	-	$15 x$
							+	$8 x$	-	$4$
							-	$8 x$	+	$4$

Step 4.2.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$36 x^{2}$	-	$15 x$	+	$4$
$2 x$	-	$1$		$72 x^{3}$	-	$66 x^{2}$	+	$23 x$	-	$4$
			-	$72 x^{3}$	+	$36 x^{2}$
					-	$30 x^{2}$	+	$23 x$
					+	$30 x^{2}$	-	$15 x$
							+	$8 x$	-	$4$
							-	$8 x$	+	$4$
										$0$

Step 4.2.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$36 x^{2} - 15 x + 4$

Step 4.2.1.6

Write $72 x^{3} - 66 x^{2} + 23 x - 4$ as a set of factors.

$(12 x + 1) ((2 x - 1) (36 x^{2} - 15 x + 4)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$(12 x + 1) (2 x - 1) (36 x^{2} - 15 x + 4) = 0$

Step 5

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

$12 x + 1 = 0$

$2 x - 1 = 0$

$36 x^{2} - 15 x + 4 = 0$

Step 6

Set $12 x + 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $12 x + 1$ equal to $0$ .

$12 x + 1 = 0$

Step 6.2

Solve $12 x + 1 = 0$ for $x$ .

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Step 6.2.1

Subtract $1$ from both sides of the equation.

$12 x = - 1$

Step 6.2.2

Divide each term in $12 x = - 1$ by $12$ and simplify.

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Step 6.2.2.1

Divide each term in $12 x = - 1$ by $12$ .

$\frac{12 x}{12} = \frac{- 1}{12}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $12$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{- 1}{12}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{12}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{12}$

Step 7

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 7.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 7.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 7.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 7.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 8

Set $36 x^{2} - 15 x + 4$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $36 x^{2} - 15 x + 4$ equal to $0$ .

$36 x^{2} - 15 x + 4 = 0$

Step 8.2

Solve $36 x^{2} - 15 x + 4 = 0$ for $x$ .

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Step 8.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 8.2.2

Substitute the values $a = 36$ , $b = - 15$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{15 \pm \sqrt{{(- 15)}^{2} - 4 \cdot (36 \cdot 4)}}{2 \cdot 36}$

Step 8.2.3

Simplify.

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Step 8.2.3.1

Simplify the numerator.

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Step 8.2.3.1.1

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 36 \cdot 4}}{2 \cdot 36}$

Step 8.2.3.1.2

Multiply $- 4 \cdot 36 \cdot 4$ .

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Step 8.2.3.1.2.1

Multiply $- 4$ by $36$ .

$x = \frac{15 \pm \sqrt{225 - 144 \cdot 4}}{2 \cdot 36}$

Step 8.2.3.1.2.2

Multiply $- 144$ by $4$ .

$x = \frac{15 \pm \sqrt{225 - 576}}{2 \cdot 36}$

Step 8.2.3.1.3

Subtract $576$ from $225$ .

$x = \frac{15 \pm \sqrt{- 351}}{2 \cdot 36}$

Step 8.2.3.1.4

Rewrite $- 351$ as $- 1 (351)$ .

$x = \frac{15 \pm \sqrt{- 1 \cdot 351}}{2 \cdot 36}$

Step 8.2.3.1.5

Rewrite $\sqrt{- 1 (351)}$ as $\sqrt{- 1} \cdot \sqrt{351}$ .

$x = \frac{15 \pm \sqrt{- 1} \cdot \sqrt{351}}{2 \cdot 36}$

Step 8.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{15 \pm i \cdot \sqrt{351}}{2 \cdot 36}$

Step 8.2.3.1.7

Rewrite $351$ as $3^{2} \cdot 39$ .

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Step 8.2.3.1.7.1

Factor $9$ out of $351$ .

$x = \frac{15 \pm i \cdot \sqrt{9 (39)}}{2 \cdot 36}$

Step 8.2.3.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{15 \pm i \cdot \sqrt{3^{2} \cdot 39}}{2 \cdot 36}$

Step 8.2.3.1.8

Pull terms out from under the radical.

$x = \frac{15 \pm i \cdot (3 \sqrt{39})}{2 \cdot 36}$

Step 8.2.3.1.9

Move $3$ to the left of $i$ .

$x = \frac{15 \pm 3 i \sqrt{39}}{2 \cdot 36}$

Step 8.2.3.2

Multiply $2$ by $36$ .

$x = \frac{15 \pm 3 i \sqrt{39}}{72}$

Step 8.2.3.3

Simplify $\frac{15 \pm 3 i \sqrt{39}}{72}$ .

$x = \frac{5 \pm i \sqrt{39}}{24}$

Step 8.2.4

The final answer is the combination of both solutions.

$x = \frac{5 + i \sqrt{39}}{24}, \frac{5 - i \sqrt{39}}{24}$

Step 9

The final solution is all the values that make $(12 x + 1) (2 x - 1) (36 x^{2} - 15 x + 4) = 0$ true.

$x = - \frac{1}{12}, \frac{1}{2}, \frac{5 + i \sqrt{39}}{24}, \frac{5 - i \sqrt{39}}{24}$