Algebra Examples

$M (s) = - 5 (s - 4) (s - 2) (s + 1) (s + 3) (s + 5)$

Step 1

Set $- 5 (s - 4) (s - 2) (s + 1) (s + 3) (s + 5)$ equal to $0$ .

$- 5 (s - 4) (s - 2) (s + 1) (s + 3) (s + 5) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Divide each term in $- 5 (s - 4) (s - 2) (s + 1) (s + 3) (s + 5) = 0$ by $- 5$ and simplify.

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Step 2.1.1

Divide each term in $- 5 (s - 4) (s - 2) (s + 1) (s + 3) (s + 5) = 0$ by $- 5$ .

$\frac{- 5 (s - 4) (s - 2) (s + 1) (s + 3) (s + 5)}{- 5} = \frac{0}{- 5}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $- 5$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{- 5 (s - 4) (s - 2) (s + 1) (s + 3) (s + 5)}{- 5} = \frac{0}{- 5}$

Step 2.1.2.1.2

Divide $((((s - 4) (s - 2)) (s + 1)) (s + 3)) (s + 5)$ by $1$ .

$((((s - 4) (s - 2)) (s + 1)) (s + 3)) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.2

Expand $(s - 4) (s - 2)$ using the FOIL Method.

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Step 2.1.2.2.1

Apply the distributive property.

$(s (s - 2) - 4 (s - 2)) (s + 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.2.2

Apply the distributive property.

$(s \cdot s + s \cdot - 2 - 4 (s - 2)) (s + 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.2.3

Apply the distributive property.

$(s \cdot s + s \cdot - 2 - 4 s - 4 \cdot - 2) (s + 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.3

Simplify and combine like terms.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Multiply $s$ by $s$ .

$(s^{2} + s \cdot - 2 - 4 s - 4 \cdot - 2) (s + 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.3.1.2

Move $- 2$ to the left of $s$ .

$(s^{2} - 2 \cdot s - 4 s - 4 \cdot - 2) (s + 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.3.1.3

Multiply $- 4$ by $- 2$ .

$(s^{2} - 2 s - 4 s + 8) (s + 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.3.2

Subtract $4 s$ from $- 2 s$ .

$(s^{2} - 6 s + 8) (s + 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.4

Expand $(s^{2} - 6 s + 8) (s + 1)$ by multiplying each term in the first expression by each term in the second expression.

$(s^{2} s + s^{2} \cdot 1 - 6 s \cdot s - 6 s \cdot 1 + 8 s + 8 \cdot 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.5

Simplify terms.

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Step 2.1.2.5.1

Simplify each term.

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Step 2.1.2.5.1.1

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

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Step 2.1.2.5.1.1.1

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

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Step 2.1.2.5.1.1.1.1

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

$(s^{2} s^{1} + s^{2} \cdot 1 - 6 s \cdot s - 6 s \cdot 1 + 8 s + 8 \cdot 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.5.1.1.1.2

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

$(s^{2 + 1} + s^{2} \cdot 1 - 6 s \cdot s - 6 s \cdot 1 + 8 s + 8 \cdot 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.5.1.1.2

Add $2$ and $1$ .

$(s^{3} + s^{2} \cdot 1 - 6 s \cdot s - 6 s \cdot 1 + 8 s + 8 \cdot 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.5.1.2

Multiply $s^{2}$ by $1$ .

$(s^{3} + s^{2} - 6 s \cdot s - 6 s \cdot 1 + 8 s + 8 \cdot 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.5.1.3

Multiply $s$ by $s$ by adding the exponents.

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Step 2.1.2.5.1.3.1

Move $s$ .

$(s^{3} + s^{2} - 6 (s \cdot s) - 6 s \cdot 1 + 8 s + 8 \cdot 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.5.1.3.2

Multiply $s$ by $s$ .

$(s^{3} + s^{2} - 6 s^{2} - 6 s \cdot 1 + 8 s + 8 \cdot 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.5.1.4

Multiply $- 6$ by $1$ .

$(s^{3} + s^{2} - 6 s^{2} - 6 s + 8 s + 8 \cdot 1) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.5.1.5

Multiply $8$ by $1$ .

$(s^{3} + s^{2} - 6 s^{2} - 6 s + 8 s + 8) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.5.2

Simplify by adding terms.

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Step 2.1.2.5.2.1

Subtract $6 s^{2}$ from $s^{2}$ .

$(s^{3} - 5 s^{2} - 6 s + 8 s + 8) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.5.2.2

Add $- 6 s$ and $8 s$ .

$(s^{3} - 5 s^{2} + 2 s + 8) (s + 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.6

Expand $(s^{3} - 5 s^{2} + 2 s + 8) (s + 3)$ by multiplying each term in the first expression by each term in the second expression.

$(s^{3} s + s^{3} \cdot 3 - 5 s^{2} s - 5 s^{2} \cdot 3 + 2 s \cdot s + 2 s \cdot 3 + 8 s + 8 \cdot 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7

Simplify terms.

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Step 2.1.2.7.1

Simplify each term.

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Step 2.1.2.7.1.1

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

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Step 2.1.2.7.1.1.1

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

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Step 2.1.2.7.1.1.1.1

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

$(s^{3} s^{1} + s^{3} \cdot 3 - 5 s^{2} s - 5 s^{2} \cdot 3 + 2 s \cdot s + 2 s \cdot 3 + 8 s + 8 \cdot 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.1.1.1.2

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

$(s^{3 + 1} + s^{3} \cdot 3 - 5 s^{2} s - 5 s^{2} \cdot 3 + 2 s \cdot s + 2 s \cdot 3 + 8 s + 8 \cdot 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.1.1.2

Add $3$ and $1$ .

$(s^{4} + s^{3} \cdot 3 - 5 s^{2} s - 5 s^{2} \cdot 3 + 2 s \cdot s + 2 s \cdot 3 + 8 s + 8 \cdot 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.1.2

Move $3$ to the left of $s^{3}$ .

$(s^{4} + 3 \cdot s^{3} - 5 s^{2} s - 5 s^{2} \cdot 3 + 2 s \cdot s + 2 s \cdot 3 + 8 s + 8 \cdot 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.1.3

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

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Step 2.1.2.7.1.3.1

Move $s$ .

$(s^{4} + 3 s^{3} - 5 (s \cdot s^{2}) - 5 s^{2} \cdot 3 + 2 s \cdot s + 2 s \cdot 3 + 8 s + 8 \cdot 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.1.3.2

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

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Step 2.1.2.7.1.3.2.1

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

$(s^{4} + 3 s^{3} - 5 (s^{1} s^{2}) - 5 s^{2} \cdot 3 + 2 s \cdot s + 2 s \cdot 3 + 8 s + 8 \cdot 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.1.3.2.2

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

$(s^{4} + 3 s^{3} - 5 s^{1 + 2} - 5 s^{2} \cdot 3 + 2 s \cdot s + 2 s \cdot 3 + 8 s + 8 \cdot 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.1.3.3

Add $1$ and $2$ .

$(s^{4} + 3 s^{3} - 5 s^{3} - 5 s^{2} \cdot 3 + 2 s \cdot s + 2 s \cdot 3 + 8 s + 8 \cdot 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.1.4

Multiply $3$ by $- 5$ .

$(s^{4} + 3 s^{3} - 5 s^{3} - 15 s^{2} + 2 s \cdot s + 2 s \cdot 3 + 8 s + 8 \cdot 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.1.5

Multiply $s$ by $s$ by adding the exponents.

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Step 2.1.2.7.1.5.1

Move $s$ .

$(s^{4} + 3 s^{3} - 5 s^{3} - 15 s^{2} + 2 (s \cdot s) + 2 s \cdot 3 + 8 s + 8 \cdot 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.1.5.2

Multiply $s$ by $s$ .

$(s^{4} + 3 s^{3} - 5 s^{3} - 15 s^{2} + 2 s^{2} + 2 s \cdot 3 + 8 s + 8 \cdot 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.1.6

Multiply $3$ by $2$ .

$(s^{4} + 3 s^{3} - 5 s^{3} - 15 s^{2} + 2 s^{2} + 6 s + 8 s + 8 \cdot 3) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.1.7

Multiply $8$ by $3$ .

$(s^{4} + 3 s^{3} - 5 s^{3} - 15 s^{2} + 2 s^{2} + 6 s + 8 s + 24) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.2

Simplify by adding terms.

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Step 2.1.2.7.2.1

Subtract $5 s^{3}$ from $3 s^{3}$ .

$(s^{4} - 2 s^{3} - 15 s^{2} + 2 s^{2} + 6 s + 8 s + 24) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.2.2

Add $- 15 s^{2}$ and $2 s^{2}$ .

$(s^{4} - 2 s^{3} - 13 s^{2} + 6 s + 8 s + 24) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.7.2.3

Add $6 s$ and $8 s$ .

$(s^{4} - 2 s^{3} - 13 s^{2} + 14 s + 24) (s + 5) = \frac{0}{- 5}$

Step 2.1.2.8

Expand $(s^{4} - 2 s^{3} - 13 s^{2} + 14 s + 24) (s + 5)$ by multiplying each term in the first expression by each term in the second expression.

$s^{4} s + s^{4} \cdot 5 - 2 s^{3} s - 2 s^{3} \cdot 5 - 13 s^{2} s - 13 s^{2} \cdot 5 + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9

Simplify terms.

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Step 2.1.2.9.1

Simplify each term.

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Step 2.1.2.9.1.1

Multiply $s^{4}$ by $s$ by adding the exponents.

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Step 2.1.2.9.1.1.1

Multiply $s^{4}$ by $s$ .

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Step 2.1.2.9.1.1.1.1

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

$s^{4} s^{1} + s^{4} \cdot 5 - 2 s^{3} s - 2 s^{3} \cdot 5 - 13 s^{2} s - 13 s^{2} \cdot 5 + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.1.1.2

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

$s^{4 + 1} + s^{4} \cdot 5 - 2 s^{3} s - 2 s^{3} \cdot 5 - 13 s^{2} s - 13 s^{2} \cdot 5 + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.1.2

Add $4$ and $1$ .

$s^{5} + s^{4} \cdot 5 - 2 s^{3} s - 2 s^{3} \cdot 5 - 13 s^{2} s - 13 s^{2} \cdot 5 + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.2

Move $5$ to the left of $s^{4}$ .

$s^{5} + 5 \cdot s^{4} - 2 s^{3} s - 2 s^{3} \cdot 5 - 13 s^{2} s - 13 s^{2} \cdot 5 + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.3

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

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Step 2.1.2.9.1.3.1

Move $s$ .

$s^{5} + 5 s^{4} - 2 (s \cdot s^{3}) - 2 s^{3} \cdot 5 - 13 s^{2} s - 13 s^{2} \cdot 5 + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.3.2

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

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Step 2.1.2.9.1.3.2.1

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

$s^{5} + 5 s^{4} - 2 (s^{1} s^{3}) - 2 s^{3} \cdot 5 - 13 s^{2} s - 13 s^{2} \cdot 5 + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.3.2.2

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

$s^{5} + 5 s^{4} - 2 s^{1 + 3} - 2 s^{3} \cdot 5 - 13 s^{2} s - 13 s^{2} \cdot 5 + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.3.3

Add $1$ and $3$ .

$s^{5} + 5 s^{4} - 2 s^{4} - 2 s^{3} \cdot 5 - 13 s^{2} s - 13 s^{2} \cdot 5 + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.4

Multiply $5$ by $- 2$ .

$s^{5} + 5 s^{4} - 2 s^{4} - 10 s^{3} - 13 s^{2} s - 13 s^{2} \cdot 5 + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.5

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

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Step 2.1.2.9.1.5.1

Move $s$ .

$s^{5} + 5 s^{4} - 2 s^{4} - 10 s^{3} - 13 (s \cdot s^{2}) - 13 s^{2} \cdot 5 + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.5.2

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

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Step 2.1.2.9.1.5.2.1

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

$s^{5} + 5 s^{4} - 2 s^{4} - 10 s^{3} - 13 (s^{1} s^{2}) - 13 s^{2} \cdot 5 + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.5.2.2

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

$s^{5} + 5 s^{4} - 2 s^{4} - 10 s^{3} - 13 s^{1 + 2} - 13 s^{2} \cdot 5 + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.5.3

Add $1$ and $2$ .

$s^{5} + 5 s^{4} - 2 s^{4} - 10 s^{3} - 13 s^{3} - 13 s^{2} \cdot 5 + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.6

Multiply $5$ by $- 13$ .

$s^{5} + 5 s^{4} - 2 s^{4} - 10 s^{3} - 13 s^{3} - 65 s^{2} + 14 s \cdot s + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.7

Multiply $s$ by $s$ by adding the exponents.

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Step 2.1.2.9.1.7.1

Move $s$ .

$s^{5} + 5 s^{4} - 2 s^{4} - 10 s^{3} - 13 s^{3} - 65 s^{2} + 14 (s \cdot s) + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.7.2

Multiply $s$ by $s$ .

$s^{5} + 5 s^{4} - 2 s^{4} - 10 s^{3} - 13 s^{3} - 65 s^{2} + 14 s^{2} + 14 s \cdot 5 + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.8

Multiply $5$ by $14$ .

$s^{5} + 5 s^{4} - 2 s^{4} - 10 s^{3} - 13 s^{3} - 65 s^{2} + 14 s^{2} + 70 s + 24 s + 24 \cdot 5 = \frac{0}{- 5}$

Step 2.1.2.9.1.9

Multiply $24$ by $5$ .

$s^{5} + 5 s^{4} - 2 s^{4} - 10 s^{3} - 13 s^{3} - 65 s^{2} + 14 s^{2} + 70 s + 24 s + 120 = \frac{0}{- 5}$

Step 2.1.2.9.2

Simplify by adding terms.

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Step 2.1.2.9.2.1

Subtract $2 s^{4}$ from $5 s^{4}$ .

$s^{5} + 3 s^{4} - 10 s^{3} - 13 s^{3} - 65 s^{2} + 14 s^{2} + 70 s + 24 s + 120 = \frac{0}{- 5}$

Step 2.1.2.9.2.2

Subtract $13 s^{3}$ from $- 10 s^{3}$ .

$s^{5} + 3 s^{4} - 23 s^{3} - 65 s^{2} + 14 s^{2} + 70 s + 24 s + 120 = \frac{0}{- 5}$

Step 2.1.2.9.2.3

Add $- 65 s^{2}$ and $14 s^{2}$ .

$s^{5} + 3 s^{4} - 23 s^{3} - 51 s^{2} + 70 s + 24 s + 120 = \frac{0}{- 5}$

Step 2.1.2.9.2.4

Add $70 s$ and $24 s$ .

$s^{5} + 3 s^{4} - 23 s^{3} - 51 s^{2} + 94 s + 120 = \frac{0}{- 5}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $- 5$ .

$s^{5} + 3 s^{4} - 23 s^{3} - 51 s^{2} + 94 s + 120 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $s^{4}$ out of $s^{5} + 3 s^{4}$ .

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Step 2.2.1.1

Factor $s^{4}$ out of $s^{5}$ .

$s^{4} s + 3 s^{4} - 23 s^{3} - 51 s^{2} + 94 s + 120 = 0$

Step 2.2.1.2

Factor $s^{4}$ out of $3 s^{4}$ .

$s^{4} s + s^{4} \cdot 3 - 23 s^{3} - 51 s^{2} + 94 s + 120 = 0$

Step 2.2.1.3

Factor $s^{4}$ out of $s^{4} s + s^{4} \cdot 3$ .

$s^{4} (s + 3) - 23 s^{3} - 51 s^{2} + 94 s + 120 = 0$

Step 2.2.2

Factor $- 23 s^{3} - 51 s^{2} + 94 s + 120$ using the rational roots test.

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Step 2.2.2.1

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.

$p = \pm 1, \pm 120, \pm 2, \pm 60, \pm 3, \pm 40, \pm 4, \pm 30, \pm 5, \pm 24, \pm 6, \pm 20, \pm 8, \pm 15, \pm 10, \pm 12$

$q = \pm 1, \pm 23$

Step 2.2.2.2

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

$\pm 1, \pm 0.04347826, \pm 120, \pm 5.2173913, \pm 2, \pm 0.08695652, \pm 60, \pm 2.60869565, \pm 3, \pm 0.13043478, \pm 40, \pm 1.73913043, \pm 4, \pm 0.17391304, \pm 30, \pm 1.30434782, \pm 5, \pm 0.2173913, \pm 24, \pm 1.04347826, \pm 6, \pm 0.26086956, \pm 20, \pm 0.86956521, \pm 8, \pm 0.34782608, \pm 15, \pm 0.65217391, \pm 10, \pm 0.4347826, \pm 12, \pm 0.52173913$

Step 2.2.2.3

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 2.2.2.3.1

Substitute $- 3$ into the polynomial.

$- 23 {(- 3)}^{3} - 51 {(- 3)}^{2} + 94 \cdot - 3 + 120$

Step 2.2.2.3.2

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

$- 23 \cdot - 27 - 51 {(- 3)}^{2} + 94 \cdot - 3 + 120$

Step 2.2.2.3.3

Multiply $- 23$ by $- 27$ .

$621 - 51 {(- 3)}^{2} + 94 \cdot - 3 + 120$

Step 2.2.2.3.4

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

$621 - 51 \cdot 9 + 94 \cdot - 3 + 120$

Step 2.2.2.3.5

Multiply $- 51$ by $9$ .

$621 - 459 + 94 \cdot - 3 + 120$

Step 2.2.2.3.6

Subtract $459$ from $621$ .

$162 + 94 \cdot - 3 + 120$

Step 2.2.2.3.7

Multiply $94$ by $- 3$ .

$162 - 282 + 120$

Step 2.2.2.3.8

Subtract $282$ from $162$ .

$- 120 + 120$

Step 2.2.2.3.9

Add $- 120$ and $120$ .

$0$

Step 2.2.2.4

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

$\frac{- 23 s^{3} - 51 s^{2} + 94 s + 120}{s + 3}$

Step 2.2.2.5

Divide $- 23 s^{3} - 51 s^{2} + 94 s + 120$ by $s + 3$ .

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Step 2.2.2.5.1

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

$s$

$3$

$23 s^{3}$

$51 s^{2}$

$94 s$

$120$

Step 2.2.2.5.2

Divide the highest order term in the dividend $- 23 s^{3}$ by the highest order term in divisor $s$ .

				-	$23 s^{2}$
	$s$	+	$3$	-	$23 s^{3}$	-	$51 s^{2}$	+	$94 s$	+	$120$

Step 2.2.2.5.3

Multiply the new quotient term by the divisor.

			-	$23 s^{2}$
$s$	+	$3$	-	$23 s^{3}$	-	$51 s^{2}$	+	$94 s$	+	$120$
			-	$23 s^{3}$	-	$69 s^{2}$

Step 2.2.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $- 23 s^{3} - 69 s^{2}$

			-	$23 s^{2}$
$s$	+	$3$	-	$23 s^{3}$	-	$51 s^{2}$	+	$94 s$	+	$120$
			+	$23 s^{3}$	+	$69 s^{2}$

Step 2.2.2.5.5

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

			-	$23 s^{2}$
$s$	+	$3$	-	$23 s^{3}$	-	$51 s^{2}$	+	$94 s$	+	$120$
			+	$23 s^{3}$	+	$69 s^{2}$
					+	$18 s^{2}$

Step 2.2.2.5.6

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

			-	$23 s^{2}$
$s$	+	$3$	-	$23 s^{3}$	-	$51 s^{2}$	+	$94 s$	+	$120$
			+	$23 s^{3}$	+	$69 s^{2}$
					+	$18 s^{2}$	+	$94 s$

Step 2.2.2.5.7

Divide the highest order term in the dividend $18 s^{2}$ by the highest order term in divisor $s$ .

			-	$23 s^{2}$	+	$18 s$
$s$	+	$3$	-	$23 s^{3}$	-	$51 s^{2}$	+	$94 s$	+	$120$
			+	$23 s^{3}$	+	$69 s^{2}$
					+	$18 s^{2}$	+	$94 s$

Step 2.2.2.5.8

Multiply the new quotient term by the divisor.

			-	$23 s^{2}$	+	$18 s$
$s$	+	$3$	-	$23 s^{3}$	-	$51 s^{2}$	+	$94 s$	+	$120$
			+	$23 s^{3}$	+	$69 s^{2}$
					+	$18 s^{2}$	+	$94 s$
					+	$18 s^{2}$	+	$54 s$

Step 2.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $18 s^{2} + 54 s$

			-	$23 s^{2}$	+	$18 s$
$s$	+	$3$	-	$23 s^{3}$	-	$51 s^{2}$	+	$94 s$	+	$120$
			+	$23 s^{3}$	+	$69 s^{2}$
					+	$18 s^{2}$	+	$94 s$
					-	$18 s^{2}$	-	$54 s$

Step 2.2.2.5.10

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

			-	$23 s^{2}$	+	$18 s$
$s$	+	$3$	-	$23 s^{3}$	-	$51 s^{2}$	+	$94 s$	+	$120$
			+	$23 s^{3}$	+	$69 s^{2}$
					+	$18 s^{2}$	+	$94 s$
					-	$18 s^{2}$	-	$54 s$
							+	$40 s$

Step 2.2.2.5.11

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

			-	$23 s^{2}$	+	$18 s$
$s$	+	$3$	-	$23 s^{3}$	-	$51 s^{2}$	+	$94 s$	+	$120$
			+	$23 s^{3}$	+	$69 s^{2}$
					+	$18 s^{2}$	+	$94 s$
					-	$18 s^{2}$	-	$54 s$
							+	$40 s$	+	$120$

Step 2.2.2.5.12

Divide the highest order term in the dividend $40 s$ by the highest order term in divisor $s$ .

			-	$23 s^{2}$	+	$18 s$	+	$40$
$s$	+	$3$	-	$23 s^{3}$	-	$51 s^{2}$	+	$94 s$	+	$120$
			+	$23 s^{3}$	+	$69 s^{2}$
					+	$18 s^{2}$	+	$94 s$
					-	$18 s^{2}$	-	$54 s$
							+	$40 s$	+	$120$

Step 2.2.2.5.13

Multiply the new quotient term by the divisor.

			-	$23 s^{2}$	+	$18 s$	+	$40$
$s$	+	$3$	-	$23 s^{3}$	-	$51 s^{2}$	+	$94 s$	+	$120$
			+	$23 s^{3}$	+	$69 s^{2}$
					+	$18 s^{2}$	+	$94 s$
					-	$18 s^{2}$	-	$54 s$
							+	$40 s$	+	$120$
							+	$40 s$	+	$120$

Step 2.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $40 s + 120$

			-	$23 s^{2}$	+	$18 s$	+	$40$
$s$	+	$3$	-	$23 s^{3}$	-	$51 s^{2}$	+	$94 s$	+	$120$
			+	$23 s^{3}$	+	$69 s^{2}$
					+	$18 s^{2}$	+	$94 s$
					-	$18 s^{2}$	-	$54 s$
							+	$40 s$	+	$120$
							-	$40 s$	-	$120$

Step 2.2.2.5.15

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

			-	$23 s^{2}$	+	$18 s$	+	$40$
$s$	+	$3$	-	$23 s^{3}$	-	$51 s^{2}$	+	$94 s$	+	$120$
			+	$23 s^{3}$	+	$69 s^{2}$
					+	$18 s^{2}$	+	$94 s$
					-	$18 s^{2}$	-	$54 s$
							+	$40 s$	+	$120$
							-	$40 s$	-	$120$
										$0$

Step 2.2.2.5.16

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

$- 23 s^{2} + 18 s + 40$

Step 2.2.2.6

Write $- 23 s^{3} - 51 s^{2} + 94 s + 120$ as a set of factors.

$s^{4} (s + 3) + (s + 3) (- 23 s^{2} + 18 s + 40) = 0$

Step 2.2.3

Factor $s + 3$ out of $s^{4} (s + 3) + (s + 3) (- 23 s^{2} + 18 s + 40)$ .

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Step 2.2.3.1

Factor $s + 3$ out of $s^{4} (s + 3)$ .

$(s + 3) s^{4} + (s + 3) (- 23 s^{2} + 18 s + 40) = 0$

Step 2.2.3.2

Factor $s + 3$ out of $(s + 3) s^{4} + (s + 3) (- 23 s^{2} + 18 s + 40)$ .

$(s + 3) (s^{4} - 23 s^{2} + 18 s + 40) = 0$

Step 2.2.4

Factor.

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Step 2.2.4.1

Rewrite $s^{4} - 23 s^{2} + 18 s + 40$ in a factored form.

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Step 2.2.4.1.1

Factor $s^{4} - 23 s^{2} + 18 s + 40$ using the rational roots test.

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Step 2.2.4.1.1.1

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.

$p = \pm 1, \pm 40, \pm 2, \pm 20, \pm 4, \pm 10, \pm 5, \pm 8$

$q = \pm 1$

Step 2.2.4.1.1.2

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

$\pm 1, \pm 40, \pm 2, \pm 20, \pm 4, \pm 10, \pm 5, \pm 8$

Step 2.2.4.1.1.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 2.2.4.1.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{4} - 23 {(- 1)}^{2} + 18 \cdot - 1 + 40$

Step 2.2.4.1.1.3.2

Raise $- 1$ to the power of $4$ .

$1 - 23 {(- 1)}^{2} + 18 \cdot - 1 + 40$

Step 2.2.4.1.1.3.3

Raise $- 1$ to the power of $2$ .

$1 - 23 \cdot 1 + 18 \cdot - 1 + 40$

Step 2.2.4.1.1.3.4

Multiply $- 23$ by $1$ .

$1 - 23 + 18 \cdot - 1 + 40$

Step 2.2.4.1.1.3.5

Subtract $23$ from $1$ .

$- 22 + 18 \cdot - 1 + 40$

Step 2.2.4.1.1.3.6

Multiply $18$ by $- 1$ .

$- 22 - 18 + 40$

Step 2.2.4.1.1.3.7

Subtract $18$ from $- 22$ .

$- 40 + 40$

Step 2.2.4.1.1.3.8

Add $- 40$ and $40$ .

$0$

Step 2.2.4.1.1.4

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

$\frac{s^{4} - 23 s^{2} + 18 s + 40}{s + 1}$

Step 2.2.4.1.1.5

Divide $s^{4} - 23 s^{2} + 18 s + 40$ by $s + 1$ .

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Step 2.2.4.1.1.5.1

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

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

Step 2.2.4.1.1.5.2

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

$s^{3}$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

Step 2.2.4.1.1.5.3

Multiply the new quotient term by the divisor.

$s^{3}$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

Step 2.2.4.1.1.5.4

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

$s^{3}$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

Step 2.2.4.1.1.5.5

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

$s^{3}$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

Step 2.2.4.1.1.5.6

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

$s^{3}$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

Step 2.2.4.1.1.5.7

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

$s^{3}$

$s^{2}$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

Step 2.2.4.1.1.5.8

Multiply the new quotient term by the divisor.

$s^{3}$

$s^{2}$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

$s^{3}$

$s^{2}$

Step 2.2.4.1.1.5.9

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

$s^{3}$

$s^{2}$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

$s^{3}$

$s^{2}$

Step 2.2.4.1.1.5.10

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

$s^{3}$

$s^{2}$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

$s^{3}$

$s^{2}$

$22 s^{2}$

Step 2.2.4.1.1.5.11

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

$s^{3}$

$s^{2}$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

$s^{3}$

$s^{2}$

$22 s^{2}$

$18 s$

Step 2.2.4.1.1.5.12

Divide the highest order term in the dividend $- 22 s^{2}$ by the highest order term in divisor $s$ .

$s^{3}$

$s^{2}$

$22 s$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

$s^{3}$

$s^{2}$

$22 s^{2}$

$18 s$

Step 2.2.4.1.1.5.13

Multiply the new quotient term by the divisor.

$s^{3}$

$s^{2}$

$22 s$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

$s^{3}$

$s^{2}$

$22 s^{2}$

$18 s$

$22 s^{2}$

$22 s$

Step 2.2.4.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 22 s^{2} - 22 s$

$s^{3}$

$s^{2}$

$22 s$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

$s^{3}$

$s^{2}$

$22 s^{2}$

$18 s$

$22 s^{2}$

$22 s$

Step 2.2.4.1.1.5.15

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

$s^{3}$

$s^{2}$

$22 s$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

$s^{3}$

$s^{2}$

$22 s^{2}$

$18 s$

$22 s^{2}$

$22 s$

$40 s$

Step 2.2.4.1.1.5.16

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

$s^{3}$

$s^{2}$

$22 s$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

$s^{3}$

$s^{2}$

$22 s^{2}$

$18 s$

$22 s^{2}$

$22 s$

$40 s$

$40$

Step 2.2.4.1.1.5.17

Divide the highest order term in the dividend $40 s$ by the highest order term in divisor $s$ .

$s^{3}$

$s^{2}$

$22 s$

$40$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

$s^{3}$

$s^{2}$

$22 s^{2}$

$18 s$

$22 s^{2}$

$22 s$

$40 s$

$40$

Step 2.2.4.1.1.5.18

Multiply the new quotient term by the divisor.

$s^{3}$

$s^{2}$

$22 s$

$40$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

$s^{3}$

$s^{2}$

$22 s^{2}$

$18 s$

$22 s^{2}$

$22 s$

$40 s$

$40$

$40 s$

$40$

Step 2.2.4.1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $40 s + 40$

$s^{3}$

$s^{2}$

$22 s$

$40$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

$s^{3}$

$s^{2}$

$22 s^{2}$

$18 s$

$22 s^{2}$

$22 s$

$40 s$

$40$

$40 s$

$40$

Step 2.2.4.1.1.5.20

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

$s^{3}$

$s^{2}$

$22 s$

$40$

$s$

$1$

$s^{4}$

$0 s^{3}$

$23 s^{2}$

$18 s$

$40$

$s^{4}$

$s^{3}$

$23 s^{2}$

$s^{3}$

$s^{2}$

$22 s^{2}$

$18 s$

$22 s^{2}$

$22 s$

$40 s$

$40$

$40 s$

$40$

$0$

Step 2.2.4.1.1.5.21

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

$s^{3} - s^{2} - 22 s + 40$

Step 2.2.4.1.1.6

Write $s^{4} - 23 s^{2} + 18 s + 40$ as a set of factors.

$(s + 3) ((s + 1) (s^{3} - s^{2} - 22 s + 40)) = 0$

Step 2.2.4.1.2

Factor $s^{3} - s^{2} - 22 s + 40$ using the rational roots test.

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Step 2.2.4.1.2.1

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.

$p = \pm 1, \pm 40, \pm 2, \pm 20, \pm 4, \pm 10, \pm 5, \pm 8$

$q = \pm 1$

Step 2.2.4.1.2.2

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

$\pm 1, \pm 40, \pm 2, \pm 20, \pm 4, \pm 10, \pm 5, \pm 8$

Step 2.2.4.1.2.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 2.2.4.1.2.3.1

Substitute $2$ into the polynomial.

$2^{3} - 2^{2} - 22 \cdot 2 + 40$

Step 2.2.4.1.2.3.2

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

$8 - 2^{2} - 22 \cdot 2 + 40$

Step 2.2.4.1.2.3.3

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

$8 - 1 \cdot 4 - 22 \cdot 2 + 40$

Step 2.2.4.1.2.3.4

Multiply $- 1$ by $4$ .

$8 - 4 - 22 \cdot 2 + 40$

Step 2.2.4.1.2.3.5

Subtract $4$ from $8$ .

$4 - 22 \cdot 2 + 40$

Step 2.2.4.1.2.3.6

Multiply $- 22$ by $2$ .

$4 - 44 + 40$

Step 2.2.4.1.2.3.7

Subtract $44$ from $4$ .

$- 40 + 40$

Step 2.2.4.1.2.3.8

Add $- 40$ and $40$ .

$0$

Step 2.2.4.1.2.4

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

$\frac{s^{3} - s^{2} - 22 s + 40}{s - 2}$

Step 2.2.4.1.2.5

Divide $s^{3} - s^{2} - 22 s + 40$ by $s - 2$ .

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Step 2.2.4.1.2.5.1

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

$s$

$2$

$s^{3}$

$s^{2}$

$22 s$

$40$

Step 2.2.4.1.2.5.2

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

					$s^{2}$
	$s$	-	$2$		$s^{3}$	-	$s^{2}$	-	$22 s$	+	$40$

Step 2.2.4.1.2.5.3

Multiply the new quotient term by the divisor.

				$s^{2}$
$s$	-	$2$		$s^{3}$	-	$s^{2}$	-	$22 s$	+	$40$
			+	$s^{3}$	-	$2 s^{2}$

Step 2.2.4.1.2.5.4

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

				$s^{2}$
$s$	-	$2$		$s^{3}$	-	$s^{2}$	-	$22 s$	+	$40$
			-	$s^{3}$	+	$2 s^{2}$

Step 2.2.4.1.2.5.5

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

				$s^{2}$
$s$	-	$2$		$s^{3}$	-	$s^{2}$	-	$22 s$	+	$40$
			-	$s^{3}$	+	$2 s^{2}$
					+	$s^{2}$

Step 2.2.4.1.2.5.6

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

				$s^{2}$
$s$	-	$2$		$s^{3}$	-	$s^{2}$	-	$22 s$	+	$40$
			-	$s^{3}$	+	$2 s^{2}$
					+	$s^{2}$	-	$22 s$

Step 2.2.4.1.2.5.7

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

				$s^{2}$	+	$s$
$s$	-	$2$		$s^{3}$	-	$s^{2}$	-	$22 s$	+	$40$
			-	$s^{3}$	+	$2 s^{2}$
					+	$s^{2}$	-	$22 s$

Step 2.2.4.1.2.5.8

Multiply the new quotient term by the divisor.

				$s^{2}$	+	$s$
$s$	-	$2$		$s^{3}$	-	$s^{2}$	-	$22 s$	+	$40$
			-	$s^{3}$	+	$2 s^{2}$
					+	$s^{2}$	-	$22 s$
					+	$s^{2}$	-	$2 s$

Step 2.2.4.1.2.5.9

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

				$s^{2}$	+	$s$
$s$	-	$2$		$s^{3}$	-	$s^{2}$	-	$22 s$	+	$40$
			-	$s^{3}$	+	$2 s^{2}$
					+	$s^{2}$	-	$22 s$
					-	$s^{2}$	+	$2 s$

Step 2.2.4.1.2.5.10

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

				$s^{2}$	+	$s$
$s$	-	$2$		$s^{3}$	-	$s^{2}$	-	$22 s$	+	$40$
			-	$s^{3}$	+	$2 s^{2}$
					+	$s^{2}$	-	$22 s$
					-	$s^{2}$	+	$2 s$
							-	$20 s$

Step 2.2.4.1.2.5.11

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

				$s^{2}$	+	$s$
$s$	-	$2$		$s^{3}$	-	$s^{2}$	-	$22 s$	+	$40$
			-	$s^{3}$	+	$2 s^{2}$
					+	$s^{2}$	-	$22 s$
					-	$s^{2}$	+	$2 s$
							-	$20 s$	+	$40$

Step 2.2.4.1.2.5.12

Divide the highest order term in the dividend $- 20 s$ by the highest order term in divisor $s$ .

				$s^{2}$	+	$s$	-	$20$
$s$	-	$2$		$s^{3}$	-	$s^{2}$	-	$22 s$	+	$40$
			-	$s^{3}$	+	$2 s^{2}$
					+	$s^{2}$	-	$22 s$
					-	$s^{2}$	+	$2 s$
							-	$20 s$	+	$40$

Step 2.2.4.1.2.5.13

Multiply the new quotient term by the divisor.

				$s^{2}$	+	$s$	-	$20$
$s$	-	$2$		$s^{3}$	-	$s^{2}$	-	$22 s$	+	$40$
			-	$s^{3}$	+	$2 s^{2}$
					+	$s^{2}$	-	$22 s$
					-	$s^{2}$	+	$2 s$
							-	$20 s$	+	$40$
							-	$20 s$	+	$40$

Step 2.2.4.1.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 s + 40$

				$s^{2}$	+	$s$	-	$20$
$s$	-	$2$		$s^{3}$	-	$s^{2}$	-	$22 s$	+	$40$
			-	$s^{3}$	+	$2 s^{2}$
					+	$s^{2}$	-	$22 s$
					-	$s^{2}$	+	$2 s$
							-	$20 s$	+	$40$
							+	$20 s$	-	$40$

Step 2.2.4.1.2.5.15

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

				$s^{2}$	+	$s$	-	$20$
$s$	-	$2$		$s^{3}$	-	$s^{2}$	-	$22 s$	+	$40$
			-	$s^{3}$	+	$2 s^{2}$
					+	$s^{2}$	-	$22 s$
					-	$s^{2}$	+	$2 s$
							-	$20 s$	+	$40$
							+	$20 s$	-	$40$
										$0$

Step 2.2.4.1.2.5.16

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

$s^{2} + s - 20$

Step 2.2.4.1.2.6

Write $s^{3} - s^{2} - 22 s + 40$ as a set of factors.

$(s + 3) ((s + 1) ((s - 2) (s^{2} + s - 20))) = 0$

Step 2.2.4.1.3

Factor $s^{2} + s - 20$ using the AC method.

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Step 2.2.4.1.3.1

Factor $s^{2} + s - 20$ using the AC method.

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Step 2.2.4.1.3.1.1

Factor $s^{2} + s - 20$ using the AC method.

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Step 2.2.4.1.3.1.1.1

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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Step 2.2.4.1.3.1.1.2

Write the factored form using these integers.

$(s + 3) ((s + 1) ((s - 2) ((s - 4) (s + 5)))) = 0$

Step 2.2.4.1.3.1.2

Remove unnecessary parentheses.

$(s + 3) ((s + 1) ((s - 2) (s - 4) (s + 5))) = 0$

Step 2.2.4.1.3.2

Remove unnecessary parentheses.

$(s + 3) ((s + 1) (s - 2) (s - 4) (s + 5)) = 0$

Step 2.2.4.2

Remove unnecessary parentheses.

$(s + 3) (s + 1) (s - 2) (s - 4) (s + 5) = 0$

Step 2.3

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

$s + 3 = 0$

$s + 1 = 0$

$s - 2 = 0$

$s - 4 = 0$

$s + 5 = 0$

Step 2.4

Set $s + 3$ equal to $0$ and solve for $s$ .

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Step 2.4.1

Set $s + 3$ equal to $0$ .

$s + 3 = 0$

Step 2.4.2

Subtract $3$ from both sides of the equation.

$s = - 3$

Step 2.5

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

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Step 2.5.1

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

$s + 1 = 0$

Step 2.5.2

Subtract $1$ from both sides of the equation.

$s = - 1$

Step 2.6

Set $s - 2$ equal to $0$ and solve for $s$ .

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Step 2.6.1

Set $s - 2$ equal to $0$ .

$s - 2 = 0$

Step 2.6.2

Add $2$ to both sides of the equation.

$s = 2$

Step 2.7

Set $s - 4$ equal to $0$ and solve for $s$ .

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Step 2.7.1

Set $s - 4$ equal to $0$ .

$s - 4 = 0$

Step 2.7.2

Add $4$ to both sides of the equation.

$s = 4$

Step 2.8

Set $s + 5$ equal to $0$ and solve for $s$ .

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Step 2.8.1

Set $s + 5$ equal to $0$ .

$s + 5 = 0$

Step 2.8.2

Subtract $5$ from both sides of the equation.

$s = - 5$

Step 2.9

The final solution is all the values that make $(s + 3) (s + 1) (s - 2) (s - 4) (s + 5) = 0$ true.

$s = - 3, - 1, 2, 4, - 5$

Step 3