Finite Math Examples

$(2 x - 5) (x - 2) (x + 4) (2 x + 7) = 91$

Step 1

Simplify $(2 x - 5) (x - 2) (x + 4) (2 x + 7)$ .

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Step 1.1

Expand $(2 x - 5) (x - 2)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$(2 x (x - 2) - 5 (x - 2)) (x + 4) (2 x + 7) = 91$

Step 1.1.2

Apply the distributive property.

$(2 x \cdot x + 2 x \cdot - 2 - 5 (x - 2)) (x + 4) (2 x + 7) = 91$

Step 1.1.3

Apply the distributive property.

$(2 x \cdot x + 2 x \cdot - 2 - 5 x - 5 \cdot - 2) (x + 4) (2 x + 7) = 91$

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

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

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Step 1.2.1.1.1

Move $x$ .

$(2 (x \cdot x) + 2 x \cdot - 2 - 5 x - 5 \cdot - 2) (x + 4) (2 x + 7) = 91$

Step 1.2.1.1.2

Multiply $x$ by $x$ .

$(2 x^{2} + 2 x \cdot - 2 - 5 x - 5 \cdot - 2) (x + 4) (2 x + 7) = 91$

Step 1.2.1.2

Multiply $- 2$ by $2$ .

$(2 x^{2} - 4 x - 5 x - 5 \cdot - 2) (x + 4) (2 x + 7) = 91$

Step 1.2.1.3

Multiply $- 5$ by $- 2$ .

$(2 x^{2} - 4 x - 5 x + 10) (x + 4) (2 x + 7) = 91$

Step 1.2.2

Subtract $5 x$ from $- 4 x$ .

$(2 x^{2} - 9 x + 10) (x + 4) (2 x + 7) = 91$

Step 1.3

Expand $(2 x^{2} - 9 x + 10) (x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$(2 x^{2} x + 2 x^{2} \cdot 4 - 9 x \cdot x - 9 x \cdot 4 + 10 x + 10 \cdot 4) (2 x + 7) = 91$

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

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

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Step 1.4.1.1.1

Move $x$ .

$(2 (x \cdot x^{2}) + 2 x^{2} \cdot 4 - 9 x \cdot x - 9 x \cdot 4 + 10 x + 10 \cdot 4) (2 x + 7) = 91$

Step 1.4.1.1.2

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

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Step 1.4.1.1.2.1

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

$(2 (x^{1} x^{2}) + 2 x^{2} \cdot 4 - 9 x \cdot x - 9 x \cdot 4 + 10 x + 10 \cdot 4) (2 x + 7) = 91$

Step 1.4.1.1.2.2

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

$(2 x^{1 + 2} + 2 x^{2} \cdot 4 - 9 x \cdot x - 9 x \cdot 4 + 10 x + 10 \cdot 4) (2 x + 7) = 91$

Step 1.4.1.1.3

Add $1$ and $2$ .

$(2 x^{3} + 2 x^{2} \cdot 4 - 9 x \cdot x - 9 x \cdot 4 + 10 x + 10 \cdot 4) (2 x + 7) = 91$

Step 1.4.1.2

Multiply $4$ by $2$ .

$(2 x^{3} + 8 x^{2} - 9 x \cdot x - 9 x \cdot 4 + 10 x + 10 \cdot 4) (2 x + 7) = 91$

Step 1.4.1.3

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

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Step 1.4.1.3.1

Move $x$ .

$(2 x^{3} + 8 x^{2} - 9 (x \cdot x) - 9 x \cdot 4 + 10 x + 10 \cdot 4) (2 x + 7) = 91$

Step 1.4.1.3.2

Multiply $x$ by $x$ .

$(2 x^{3} + 8 x^{2} - 9 x^{2} - 9 x \cdot 4 + 10 x + 10 \cdot 4) (2 x + 7) = 91$

Step 1.4.1.4

Multiply $4$ by $- 9$ .

$(2 x^{3} + 8 x^{2} - 9 x^{2} - 36 x + 10 x + 10 \cdot 4) (2 x + 7) = 91$

Step 1.4.1.5

Multiply $10$ by $4$ .

$(2 x^{3} + 8 x^{2} - 9 x^{2} - 36 x + 10 x + 40) (2 x + 7) = 91$

Step 1.4.2

Simplify by adding terms.

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Step 1.4.2.1

Subtract $9 x^{2}$ from $8 x^{2}$ .

$(2 x^{3} - x^{2} - 36 x + 10 x + 40) (2 x + 7) = 91$

Step 1.4.2.2

Add $- 36 x$ and $10 x$ .

$(2 x^{3} - x^{2} - 26 x + 40) (2 x + 7) = 91$

Step 1.5

Expand $(2 x^{3} - x^{2} - 26 x + 40) (2 x + 7)$ by multiplying each term in the first expression by each term in the second expression.

$2 x^{3} (2 x) + 2 x^{3} \cdot 7 - x^{2} (2 x) - x^{2} \cdot 7 - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

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.

$2 \cdot 2 x^{3} x + 2 x^{3} \cdot 7 - x^{2} (2 x) - x^{2} \cdot 7 - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

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 $x$ .

$2 \cdot 2 (x \cdot x^{3}) + 2 x^{3} \cdot 7 - x^{2} (2 x) - x^{2} \cdot 7 - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

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 $1$ .

$2 \cdot 2 (x^{1} x^{3}) + 2 x^{3} \cdot 7 - x^{2} (2 x) - x^{2} \cdot 7 - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.2.2.2

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

$2 \cdot 2 x^{1 + 3} + 2 x^{3} \cdot 7 - x^{2} (2 x) - x^{2} \cdot 7 - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.2.3

Add $1$ and $3$ .

$2 \cdot 2 x^{4} + 2 x^{3} \cdot 7 - x^{2} (2 x) - x^{2} \cdot 7 - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.3

Multiply $2$ by $2$ .

$4 x^{4} + 2 x^{3} \cdot 7 - x^{2} (2 x) - x^{2} \cdot 7 - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.4

Multiply $7$ by $2$ .

$4 x^{4} + 14 x^{3} - x^{2} (2 x) - x^{2} \cdot 7 - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.5

Rewrite using the commutative property of multiplication.

$4 x^{4} + 14 x^{3} - 1 \cdot 2 x^{2} x - x^{2} \cdot 7 - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

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 $x$ .

$4 x^{4} + 14 x^{3} - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot 7 - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

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 $1$ .

$4 x^{4} + 14 x^{3} - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot 7 - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.6.2.2

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

$4 x^{4} + 14 x^{3} - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot 7 - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.6.3

Add $1$ and $2$ .

$4 x^{4} + 14 x^{3} - 1 \cdot 2 x^{3} - x^{2} \cdot 7 - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.7

Multiply $- 1$ by $2$ .

$4 x^{4} + 14 x^{3} - 2 x^{3} - x^{2} \cdot 7 - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.8

Multiply $7$ by $- 1$ .

$4 x^{4} + 14 x^{3} - 2 x^{3} - 7 x^{2} - 26 x (2 x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.9

Rewrite using the commutative property of multiplication.

$4 x^{4} + 14 x^{3} - 2 x^{3} - 7 x^{2} - 26 \cdot 2 x \cdot x - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.10

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

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Step 1.6.1.10.1

Move $x$ .

$4 x^{4} + 14 x^{3} - 2 x^{3} - 7 x^{2} - 26 \cdot 2 (x \cdot x) - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.10.2

Multiply $x$ by $x$ .

$4 x^{4} + 14 x^{3} - 2 x^{3} - 7 x^{2} - 26 \cdot 2 x^{2} - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.11

Multiply $- 26$ by $2$ .

$4 x^{4} + 14 x^{3} - 2 x^{3} - 7 x^{2} - 52 x^{2} - 26 x \cdot 7 + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.12

Multiply $7$ by $- 26$ .

$4 x^{4} + 14 x^{3} - 2 x^{3} - 7 x^{2} - 52 x^{2} - 182 x + 40 (2 x) + 40 \cdot 7 = 91$

Step 1.6.1.13

Multiply $2$ by $40$ .

$4 x^{4} + 14 x^{3} - 2 x^{3} - 7 x^{2} - 52 x^{2} - 182 x + 80 x + 40 \cdot 7 = 91$

Step 1.6.1.14

Multiply $40$ by $7$ .

$4 x^{4} + 14 x^{3} - 2 x^{3} - 7 x^{2} - 52 x^{2} - 182 x + 80 x + 280 = 91$

Step 1.6.2

Simplify by adding terms.

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Step 1.6.2.1

Subtract $2 x^{3}$ from $14 x^{3}$ .

$4 x^{4} + 12 x^{3} - 7 x^{2} - 52 x^{2} - 182 x + 80 x + 280 = 91$

Step 1.6.2.2

Subtract $52 x^{2}$ from $- 7 x^{2}$ .

$4 x^{4} + 12 x^{3} - 59 x^{2} - 182 x + 80 x + 280 = 91$

Step 1.6.2.3

Add $- 182 x$ and $80 x$ .

$4 x^{4} + 12 x^{3} - 59 x^{2} - 102 x + 280 = 91$

Step 2

Subtract $91$ from both sides of the equation.

$4 x^{4} + 12 x^{3} - 59 x^{2} - 102 x + 280 - 91 = 0$

Step 3

Subtract $91$ from $280$ .

$4 x^{4} + 12 x^{3} - 59 x^{2} - 102 x + 189 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $4 x^{4} + 12 x^{3} - 59 x^{2} - 102 x + 189$ 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.

$p = \pm 1, \pm 189, \pm 3, \pm 63, \pm 7, \pm 27, \pm 9, \pm 21$

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

Step 4.1.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 189, \pm 47.25, \pm 94.5, \pm 3, \pm 0.75, \pm 1.5, \pm 63, \pm 15.75, \pm 31.5, \pm 7, \pm 1.75, \pm 3.5, \pm 27, \pm 6.75, \pm 13.5, \pm 9, \pm 2.25, \pm 4.5, \pm 21, \pm 5.25, \pm 10.5$

Step 4.1.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

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Step 4.1.3.1

Substitute $3$ into the polynomial.

$4 \cdot 3^{4} + 12 \cdot 3^{3} - 59 \cdot 3^{2} - 102 \cdot 3 + 189$

Step 4.1.3.2

Raise $3$ to the power of $4$ .

$4 \cdot 81 + 12 \cdot 3^{3} - 59 \cdot 3^{2} - 102 \cdot 3 + 189$

Step 4.1.3.3

Multiply $4$ by $81$ .

$324 + 12 \cdot 3^{3} - 59 \cdot 3^{2} - 102 \cdot 3 + 189$

Step 4.1.3.4

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

$324 + 12 \cdot 27 - 59 \cdot 3^{2} - 102 \cdot 3 + 189$

Step 4.1.3.5

Multiply $12$ by $27$ .

$324 + 324 - 59 \cdot 3^{2} - 102 \cdot 3 + 189$

Step 4.1.3.6

Add $324$ and $324$ .

$648 - 59 \cdot 3^{2} - 102 \cdot 3 + 189$

Step 4.1.3.7

Raise $3$ to the power of $2$ .

$648 - 59 \cdot 9 - 102 \cdot 3 + 189$

Step 4.1.3.8

Multiply $- 59$ by $9$ .

$648 - 531 - 102 \cdot 3 + 189$

Step 4.1.3.9

Subtract $531$ from $648$ .

$117 - 102 \cdot 3 + 189$

Step 4.1.3.10

Multiply $- 102$ by $3$ .

$117 - 306 + 189$

Step 4.1.3.11

Subtract $306$ from $117$ .

$- 189 + 189$

Step 4.1.3.12

Add $- 189$ and $189$ .

$0$

Step 4.1.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{4} + 12 x^{3} - 59 x^{2} - 102 x + 189}{x - 3}$

Step 4.1.5

Divide $4 x^{4} + 12 x^{3} - 59 x^{2} - 102 x + 189$ by $x - 3$ .

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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 $0$ .

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

Step 4.1.5.2

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

$4 x^{3}$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

Step 4.1.5.3

Multiply the new quotient term by the divisor.

$4 x^{3}$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

Step 4.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{4} - 12 x^{3}$

$4 x^{3}$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

Step 4.1.5.5

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

$4 x^{3}$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

Step 4.1.5.6

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

$4 x^{3}$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

Step 4.1.5.7

Divide the highest order term in the dividend $24 x^{3}$ by the highest order term in divisor $x$ .

$4 x^{3}$

$24 x^{2}$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

Step 4.1.5.8

Multiply the new quotient term by the divisor.

$4 x^{3}$

$24 x^{2}$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

$24 x^{3}$

$72 x^{2}$

Step 4.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $24 x^{3} - 72 x^{2}$

$4 x^{3}$

$24 x^{2}$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

$24 x^{3}$

$72 x^{2}$

Step 4.1.5.10

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

$4 x^{3}$

$24 x^{2}$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

$24 x^{3}$

$72 x^{2}$

$13 x^{2}$

Step 4.1.5.11

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

$4 x^{3}$

$24 x^{2}$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

$24 x^{3}$

$72 x^{2}$

$13 x^{2}$

$102 x$

Step 4.1.5.12

Divide the highest order term in the dividend $13 x^{2}$ by the highest order term in divisor $x$ .

$4 x^{3}$

$24 x^{2}$

$13 x$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

$24 x^{3}$

$72 x^{2}$

$13 x^{2}$

$102 x$

Step 4.1.5.13

Multiply the new quotient term by the divisor.

$4 x^{3}$

$24 x^{2}$

$13 x$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

$24 x^{3}$

$72 x^{2}$

$13 x^{2}$

$102 x$

$13 x^{2}$

$39 x$

Step 4.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $13 x^{2} - 39 x$

$4 x^{3}$

$24 x^{2}$

$13 x$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

$24 x^{3}$

$72 x^{2}$

$13 x^{2}$

$102 x$

$13 x^{2}$

$39 x$

Step 4.1.5.15

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

$4 x^{3}$

$24 x^{2}$

$13 x$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

$24 x^{3}$

$72 x^{2}$

$13 x^{2}$

$102 x$

$13 x^{2}$

$39 x$

$63 x$

Step 4.1.5.16

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

$4 x^{3}$

$24 x^{2}$

$13 x$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

$24 x^{3}$

$72 x^{2}$

$13 x^{2}$

$102 x$

$13 x^{2}$

$39 x$

$63 x$

$189$

Step 4.1.5.17

Divide the highest order term in the dividend $- 63 x$ by the highest order term in divisor $x$ .

$4 x^{3}$

$24 x^{2}$

$13 x$

$63$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

$24 x^{3}$

$72 x^{2}$

$13 x^{2}$

$102 x$

$13 x^{2}$

$39 x$

$63 x$

$189$

Step 4.1.5.18

Multiply the new quotient term by the divisor.

$4 x^{3}$

$24 x^{2}$

$13 x$

$63$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

$24 x^{3}$

$72 x^{2}$

$13 x^{2}$

$102 x$

$13 x^{2}$

$39 x$

$63 x$

$189$

$63 x$

$189$

Step 4.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 63 x + 189$

$4 x^{3}$

$24 x^{2}$

$13 x$

$63$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

$24 x^{3}$

$72 x^{2}$

$13 x^{2}$

$102 x$

$13 x^{2}$

$39 x$

$63 x$

$189$

$63 x$

$189$

Step 4.1.5.20

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

$4 x^{3}$

$24 x^{2}$

$13 x$

$63$

$x$

$3$

$4 x^{4}$

$12 x^{3}$

$59 x^{2}$

$102 x$

$189$

$4 x^{4}$

$12 x^{3}$

$24 x^{3}$

$59 x^{2}$

$24 x^{3}$

$72 x^{2}$

$13 x^{2}$

$102 x$

$13 x^{2}$

$39 x$

$63 x$

$189$

$63 x$

$189$

$0$

Step 4.1.5.21

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

$4 x^{3} + 24 x^{2} + 13 x - 63$

Step 4.1.6

Write $4 x^{4} + 12 x^{3} - 59 x^{2} - 102 x + 189$ as a set of factors.

$(x - 3) (4 x^{3} + 24 x^{2} + 13 x - 63) = 0$

Step 4.2

Factor $4 x^{3} + 24 x^{2} + 13 x - 63$ using the rational roots test.

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Step 4.2.1

Factor $4 x^{3} + 24 x^{2} + 13 x - 63$ 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.

$p = \pm 1, \pm 63, \pm 3, \pm 21, \pm 7, \pm 9$

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

Step 4.2.1.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 63, \pm 15.75, \pm 31.5, \pm 3, \pm 0.75, \pm 1.5, \pm 21, \pm 5.25, \pm 10.5, \pm 7, \pm 1.75, \pm 3.5, \pm 9, \pm 2.25, \pm 4.5$

Step 4.2.1.3

Substitute $- 4.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 4.5$ is a root of the polynomial.

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Step 4.2.1.3.1

Substitute $- 4.5$ into the polynomial.

$4 {(- 4.5)}^{3} + 24 {(- 4.5)}^{2} + 13 \cdot - 4.5 - 63$

Step 4.2.1.3.2

Raise $- 4.5$ to the power of $3$ .

$4 \cdot - 91.125 + 24 {(- 4.5)}^{2} + 13 \cdot - 4.5 - 63$

Step 4.2.1.3.3

Multiply $4$ by $- 91.125$ .

$- 364.5 + 24 {(- 4.5)}^{2} + 13 \cdot - 4.5 - 63$

Step 4.2.1.3.4

Raise $- 4.5$ to the power of $2$ .

$- 364.5 + 24 \cdot 20.25 + 13 \cdot - 4.5 - 63$

Step 4.2.1.3.5

Multiply $24$ by $20.25$ .

$- 364.5 + 486 + 13 \cdot - 4.5 - 63$

Step 4.2.1.3.6

Add $- 364.5$ and $486$ .

$121.5 + 13 \cdot - 4.5 - 63$

Step 4.2.1.3.7

Multiply $13$ by $- 4.5$ .

$121.5 - 58.5 - 63$

Step 4.2.1.3.8

Subtract $58.5$ from $121.5$ .

$63 - 63$

Step 4.2.1.3.9

Subtract $63$ from $63$ .

$0$

Step 4.2.1.4

Since $- 4.5$ is a known root, divide the polynomial by $2 x + 9$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} + 24 x^{2} + 13 x - 63}{2 x + 9}$

Step 4.2.1.5

Divide $4 x^{3} + 24 x^{2} + 13 x - 63$ by $2 x + 9$ .

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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 $0$ .

$2 x$

$9$

$4 x^{3}$

$24 x^{2}$

$13 x$

$63$

Step 4.2.1.5.2

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $2 x$ .

					$2 x^{2}$
	$2 x$	+	$9$		$4 x^{3}$	+	$24 x^{2}$	+	$13 x$	-	$63$

Step 4.2.1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	+	$9$		$4 x^{3}$	+	$24 x^{2}$	+	$13 x$	-	$63$
			+	$4 x^{3}$	+	$18 x^{2}$

Step 4.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} + 18 x^{2}$

				$2 x^{2}$
$2 x$	+	$9$		$4 x^{3}$	+	$24 x^{2}$	+	$13 x$	-	$63$
			-	$4 x^{3}$	-	$18 x^{2}$

Step 4.2.1.5.5

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

				$2 x^{2}$
$2 x$	+	$9$		$4 x^{3}$	+	$24 x^{2}$	+	$13 x$	-	$63$
			-	$4 x^{3}$	-	$18 x^{2}$
					+	$6 x^{2}$

Step 4.2.1.5.6

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

				$2 x^{2}$
$2 x$	+	$9$		$4 x^{3}$	+	$24 x^{2}$	+	$13 x$	-	$63$
			-	$4 x^{3}$	-	$18 x^{2}$
					+	$6 x^{2}$	+	$13 x$

Step 4.2.1.5.7

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	+	$3 x$
$2 x$	+	$9$		$4 x^{3}$	+	$24 x^{2}$	+	$13 x$	-	$63$
			-	$4 x^{3}$	-	$18 x^{2}$
					+	$6 x^{2}$	+	$13 x$

Step 4.2.1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$3 x$
$2 x$	+	$9$		$4 x^{3}$	+	$24 x^{2}$	+	$13 x$	-	$63$
			-	$4 x^{3}$	-	$18 x^{2}$
					+	$6 x^{2}$	+	$13 x$
					+	$6 x^{2}$	+	$27 x$

Step 4.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} + 27 x$

				$2 x^{2}$	+	$3 x$
$2 x$	+	$9$		$4 x^{3}$	+	$24 x^{2}$	+	$13 x$	-	$63$
			-	$4 x^{3}$	-	$18 x^{2}$
					+	$6 x^{2}$	+	$13 x$
					-	$6 x^{2}$	-	$27 x$

Step 4.2.1.5.10

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

				$2 x^{2}$	+	$3 x$
$2 x$	+	$9$		$4 x^{3}$	+	$24 x^{2}$	+	$13 x$	-	$63$
			-	$4 x^{3}$	-	$18 x^{2}$
					+	$6 x^{2}$	+	$13 x$
					-	$6 x^{2}$	-	$27 x$
							-	$14 x$

Step 4.2.1.5.11

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

				$2 x^{2}$	+	$3 x$
$2 x$	+	$9$		$4 x^{3}$	+	$24 x^{2}$	+	$13 x$	-	$63$
			-	$4 x^{3}$	-	$18 x^{2}$
					+	$6 x^{2}$	+	$13 x$
					-	$6 x^{2}$	-	$27 x$
							-	$14 x$	-	$63$

Step 4.2.1.5.12

Divide the highest order term in the dividend $- 14 x$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	+	$3 x$	-	$7$
$2 x$	+	$9$		$4 x^{3}$	+	$24 x^{2}$	+	$13 x$	-	$63$
			-	$4 x^{3}$	-	$18 x^{2}$
					+	$6 x^{2}$	+	$13 x$
					-	$6 x^{2}$	-	$27 x$
							-	$14 x$	-	$63$

Step 4.2.1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$3 x$	-	$7$
$2 x$	+	$9$		$4 x^{3}$	+	$24 x^{2}$	+	$13 x$	-	$63$
			-	$4 x^{3}$	-	$18 x^{2}$
					+	$6 x^{2}$	+	$13 x$
					-	$6 x^{2}$	-	$27 x$
							-	$14 x$	-	$63$
							-	$14 x$	-	$63$

Step 4.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 14 x - 63$

				$2 x^{2}$	+	$3 x$	-	$7$
$2 x$	+	$9$		$4 x^{3}$	+	$24 x^{2}$	+	$13 x$	-	$63$
			-	$4 x^{3}$	-	$18 x^{2}$
					+	$6 x^{2}$	+	$13 x$
					-	$6 x^{2}$	-	$27 x$
							-	$14 x$	-	$63$
							+	$14 x$	+	$63$

Step 4.2.1.5.15

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

				$2 x^{2}$	+	$3 x$	-	$7$
$2 x$	+	$9$		$4 x^{3}$	+	$24 x^{2}$	+	$13 x$	-	$63$
			-	$4 x^{3}$	-	$18 x^{2}$
					+	$6 x^{2}$	+	$13 x$
					-	$6 x^{2}$	-	$27 x$
							-	$14 x$	-	$63$
							+	$14 x$	+	$63$
										$0$

Step 4.2.1.5.16

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

$2 x^{2} + 3 x - 7$

Step 4.2.1.6

Write $4 x^{3} + 24 x^{2} + 13 x - 63$ as a set of factors.

$(x - 3) ((2 x + 9) (2 x^{2} + 3 x - 7)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$(x - 3) (2 x + 9) (2 x^{2} + 3 x - 7) = 0$

Step 5

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

$x - 3 = 0$

$2 x + 9 = 0$

$2 x^{2} + 3 x - 7 = 0$

Step 6

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

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Step 6.1

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

$x - 3 = 0$

Step 6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 7

Set $2 x + 9$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $2 x + 9$ equal to $0$ .

$2 x + 9 = 0$

Step 7.2

Solve $2 x + 9 = 0$ for $x$ .

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Step 7.2.1

Subtract $9$ from both sides of the equation.

$2 x = - 9$

Step 7.2.2

Divide each term in $2 x = - 9$ by $2$ and simplify.

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Step 7.2.2.1

Divide each term in $2 x = - 9$ by $2$ .

$\frac{2 x}{2} = \frac{- 9}{2}$

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.

$\frac{2 x}{2} = \frac{- 9}{2}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{2}$

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{9}{2}$

Step 8

Set $2 x^{2} + 3 x - 7$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $2 x^{2} + 3 x - 7$ equal to $0$ .

$2 x^{2} + 3 x - 7 = 0$

Step 8.2

Solve $2 x^{2} + 3 x - 7 = 0$ 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 $a = 2$ , $b = 3$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (2 \cdot - 7)}}{2 \cdot 2}$

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 $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 7}}{2 \cdot 2}$

Step 8.2.3.1.2

Multiply $- 4 \cdot 2 \cdot - 7$ .

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Step 8.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 7}}{2 \cdot 2}$

Step 8.2.3.1.2.2

Multiply $- 8$ by $- 7$ .

$x = \frac{- 3 \pm \sqrt{9 + 56}}{2 \cdot 2}$

Step 8.2.3.1.3

Add $9$ and $56$ .

$x = \frac{- 3 \pm \sqrt{65}}{2 \cdot 2}$

Step 8.2.3.2

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm \sqrt{65}}{4}$

Step 8.2.4

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{65}}{4}, - \frac{3 + \sqrt{65}}{4}$

Step 9

The final solution is all the values that make $(x - 3) (2 x + 9) (2 x^{2} + 3 x - 7) = 0$ true.

$x = 3, - \frac{9}{2}, - \frac{3 - \sqrt{65}}{4}, - \frac{3 + \sqrt{65}}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = 3, - \frac{9}{2}, - \frac{3 - \sqrt{65}}{4}, - \frac{3 + \sqrt{65}}{4}$

Decimal Form:

$x = 3, - 4.5, 1.26556443 \dots, - 2.76556443 \dots$

Mixed Number Form:

$x = 3, - 4 \frac{1}{2}, - \frac{3 - \sqrt{65}}{4}, - \frac{3 + \sqrt{65}}{4}$