Algebra Examples

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

Step 1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$m (s) = (- 5 s - 5 \cdot - 4) (s - 2) (s + 1) (s + 3) (s + 5)$

Step 1.1.2

Multiply $- 5$ by $- 4$ .

$m (s) = (- 5 s + 20) (s - 2) (s + 1) (s + 3) (s + 5)$

Step 1.2

Expand $(- 5 s + 20) (s - 2)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$m (s) = (- 5 s (s - 2) + 20 (s - 2)) (s + 1) (s + 3) (s + 5)$

Step 1.2.2

Apply the distributive property.

$m (s) = (- 5 s \cdot s - 5 s \cdot - 2 + 20 (s - 2)) (s + 1) (s + 3) (s + 5)$

Step 1.2.3

Apply the distributive property.

$m (s) = (- 5 s \cdot s - 5 s \cdot - 2 + 20 s + 20 \cdot - 2) (s + 1) (s + 3) (s + 5)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

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Step 1.3.1.1.1

Move $s$ .

$m (s) = (- 5 (s \cdot s) - 5 s \cdot - 2 + 20 s + 20 \cdot - 2) (s + 1) (s + 3) (s + 5)$

Step 1.3.1.1.2

Multiply $s$ by $s$ .

$m (s) = (- 5 s^{2} - 5 s \cdot - 2 + 20 s + 20 \cdot - 2) (s + 1) (s + 3) (s + 5)$

Step 1.3.1.2

Multiply $- 2$ by $- 5$ .

$m (s) = (- 5 s^{2} + 10 s + 20 s + 20 \cdot - 2) (s + 1) (s + 3) (s + 5)$

Step 1.3.1.3

Multiply $20$ by $- 2$ .

$m (s) = (- 5 s^{2} + 10 s + 20 s - 40) (s + 1) (s + 3) (s + 5)$

Step 1.3.2

Add $10 s$ and $20 s$ .

$m (s) = (- 5 s^{2} + 30 s - 40) (s + 1) (s + 3) (s + 5)$

Step 1.4

Expand $(- 5 s^{2} + 30 s - 40) (s + 1)$ by multiplying each term in the first expression by each term in the second expression.

$m (s) = (- 5 s^{2} s - 5 s^{2} \cdot 1 + 30 s \cdot s + 30 s \cdot 1 - 40 s - 40 \cdot 1) (s + 3) (s + 5)$

Step 1.5

Simplify terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

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

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Step 1.5.1.1.1

Move $s$ .

$m (s) = (- 5 (s \cdot s^{2}) - 5 s^{2} \cdot 1 + 30 s \cdot s + 30 s \cdot 1 - 40 s - 40 \cdot 1) (s + 3) (s + 5)$

Step 1.5.1.1.2

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

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Step 1.5.1.1.2.1

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

$m (s) = (- 5 (s \cdot s^{2}) - 5 s^{2} \cdot 1 + 30 s \cdot s + 30 s \cdot 1 - 40 s - 40 \cdot 1) (s + 3) (s + 5)$

Step 1.5.1.1.2.2

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

$m (s) = (- 5 s^{1 + 2} - 5 s^{2} \cdot 1 + 30 s \cdot s + 30 s \cdot 1 - 40 s - 40 \cdot 1) (s + 3) (s + 5)$

Step 1.5.1.1.3

Add $1$ and $2$ .

$m (s) = (- 5 s^{3} - 5 s^{2} \cdot 1 + 30 s \cdot s + 30 s \cdot 1 - 40 s - 40 \cdot 1) (s + 3) (s + 5)$

Step 1.5.1.2

Multiply $- 5$ by $1$ .

$m (s) = (- 5 s^{3} - 5 s^{2} + 30 s \cdot s + 30 s \cdot 1 - 40 s - 40 \cdot 1) (s + 3) (s + 5)$

Step 1.5.1.3

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

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Step 1.5.1.3.1

Move $s$ .

$m (s) = (- 5 s^{3} - 5 s^{2} + 30 (s \cdot s) + 30 s \cdot 1 - 40 s - 40 \cdot 1) (s + 3) (s + 5)$

Step 1.5.1.3.2

Multiply $s$ by $s$ .

$m (s) = (- 5 s^{3} - 5 s^{2} + 30 s^{2} + 30 s \cdot 1 - 40 s - 40 \cdot 1) (s + 3) (s + 5)$

Step 1.5.1.4

Multiply $30$ by $1$ .

$m (s) = (- 5 s^{3} - 5 s^{2} + 30 s^{2} + 30 s - 40 s - 40 \cdot 1) (s + 3) (s + 5)$

Step 1.5.1.5

Multiply $- 40$ by $1$ .

$m (s) = (- 5 s^{3} - 5 s^{2} + 30 s^{2} + 30 s - 40 s - 40) (s + 3) (s + 5)$

Step 1.5.2

Simplify by adding terms.

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Step 1.5.2.1

Add $- 5 s^{2}$ and $30 s^{2}$ .

$m (s) = (- 5 s^{3} + 25 s^{2} + 30 s - 40 s - 40) (s + 3) (s + 5)$

Step 1.5.2.2

Subtract $40 s$ from $30 s$ .

$m (s) = (- 5 s^{3} + 25 s^{2} - 10 s - 40) (s + 3) (s + 5)$

Step 1.6

Expand $(- 5 s^{3} + 25 s^{2} - 10 s - 40) (s + 3)$ by multiplying each term in the first expression by each term in the second expression.

$m (s) = (- 5 s^{3} s - 5 s^{3} \cdot 3 + 25 s^{2} s + 25 s^{2} \cdot 3 - 10 s \cdot s - 10 s \cdot 3 - 40 s - 40 \cdot 3) (s + 5)$

Step 1.7

Simplify terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

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

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Step 1.7.1.1.1

Move $s$ .

$m (s) = (- 5 (s \cdot s^{3}) - 5 s^{3} \cdot 3 + 25 s^{2} s + 25 s^{2} \cdot 3 - 10 s \cdot s - 10 s \cdot 3 - 40 s - 40 \cdot 3) (s + 5)$

Step 1.7.1.1.2

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

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Step 1.7.1.1.2.1

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

$m (s) = (- 5 (s \cdot s^{3}) - 5 s^{3} \cdot 3 + 25 s^{2} s + 25 s^{2} \cdot 3 - 10 s \cdot s - 10 s \cdot 3 - 40 s - 40 \cdot 3) (s + 5)$

Step 1.7.1.1.2.2

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

$m (s) = (- 5 s^{1 + 3} - 5 s^{3} \cdot 3 + 25 s^{2} s + 25 s^{2} \cdot 3 - 10 s \cdot s - 10 s \cdot 3 - 40 s - 40 \cdot 3) (s + 5)$

Step 1.7.1.1.3

Add $1$ and $3$ .

$m (s) = (- 5 s^{4} - 5 s^{3} \cdot 3 + 25 s^{2} s + 25 s^{2} \cdot 3 - 10 s \cdot s - 10 s \cdot 3 - 40 s - 40 \cdot 3) (s + 5)$

Step 1.7.1.2

Multiply $3$ by $- 5$ .

$m (s) = (- 5 s^{4} - 15 s^{3} + 25 s^{2} s + 25 s^{2} \cdot 3 - 10 s \cdot s - 10 s \cdot 3 - 40 s - 40 \cdot 3) (s + 5)$

Step 1.7.1.3

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

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Step 1.7.1.3.1

Move $s$ .

$m (s) = (- 5 s^{4} - 15 s^{3} + 25 (s \cdot s^{2}) + 25 s^{2} \cdot 3 - 10 s \cdot s - 10 s \cdot 3 - 40 s - 40 \cdot 3) (s + 5)$

Step 1.7.1.3.2

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

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Step 1.7.1.3.2.1

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

$m (s) = (- 5 s^{4} - 15 s^{3} + 25 (s \cdot s^{2}) + 25 s^{2} \cdot 3 - 10 s \cdot s - 10 s \cdot 3 - 40 s - 40 \cdot 3) (s + 5)$

Step 1.7.1.3.2.2

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

$m (s) = (- 5 s^{4} - 15 s^{3} + 25 s^{1 + 2} + 25 s^{2} \cdot 3 - 10 s \cdot s - 10 s \cdot 3 - 40 s - 40 \cdot 3) (s + 5)$

Step 1.7.1.3.3

Add $1$ and $2$ .

$m (s) = (- 5 s^{4} - 15 s^{3} + 25 s^{3} + 25 s^{2} \cdot 3 - 10 s \cdot s - 10 s \cdot 3 - 40 s - 40 \cdot 3) (s + 5)$

Step 1.7.1.4

Multiply $3$ by $25$ .

$m (s) = (- 5 s^{4} - 15 s^{3} + 25 s^{3} + 75 s^{2} - 10 s \cdot s - 10 s \cdot 3 - 40 s - 40 \cdot 3) (s + 5)$

Step 1.7.1.5

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

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Step 1.7.1.5.1

Move $s$ .

$m (s) = (- 5 s^{4} - 15 s^{3} + 25 s^{3} + 75 s^{2} - 10 (s \cdot s) - 10 s \cdot 3 - 40 s - 40 \cdot 3) (s + 5)$

Step 1.7.1.5.2

Multiply $s$ by $s$ .

$m (s) = (- 5 s^{4} - 15 s^{3} + 25 s^{3} + 75 s^{2} - 10 s^{2} - 10 s \cdot 3 - 40 s - 40 \cdot 3) (s + 5)$

Step 1.7.1.6

Multiply $3$ by $- 10$ .

$m (s) = (- 5 s^{4} - 15 s^{3} + 25 s^{3} + 75 s^{2} - 10 s^{2} - 30 s - 40 s - 40 \cdot 3) (s + 5)$

Step 1.7.1.7

Multiply $- 40$ by $3$ .

$m (s) = (- 5 s^{4} - 15 s^{3} + 25 s^{3} + 75 s^{2} - 10 s^{2} - 30 s - 40 s - 120) (s + 5)$

Step 1.7.2

Simplify by adding terms.

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Step 1.7.2.1

Add $- 15 s^{3}$ and $25 s^{3}$ .

$m (s) = (- 5 s^{4} + 10 s^{3} + 75 s^{2} - 10 s^{2} - 30 s - 40 s - 120) (s + 5)$

Step 1.7.2.2

Subtract $10 s^{2}$ from $75 s^{2}$ .

$m (s) = (- 5 s^{4} + 10 s^{3} + 65 s^{2} - 30 s - 40 s - 120) (s + 5)$

Step 1.7.2.3

Subtract $40 s$ from $- 30 s$ .

$m (s) = (- 5 s^{4} + 10 s^{3} + 65 s^{2} - 70 s - 120) (s + 5)$

Step 1.8

Expand $(- 5 s^{4} + 10 s^{3} + 65 s^{2} - 70 s - 120) (s + 5)$ by multiplying each term in the first expression by each term in the second expression.

$m (s) = - 5 s^{4} s - 5 s^{4} \cdot 5 + 10 s^{3} s + 10 s^{3} \cdot 5 + 65 s^{2} s + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9

Simplify terms.

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

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

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Step 1.9.1.1.1

Move $s$ .

$m (s) = - 5 (s \cdot s^{4}) - 5 s^{4} \cdot 5 + 10 s^{3} s + 10 s^{3} \cdot 5 + 65 s^{2} s + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.1.2

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

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Step 1.9.1.1.2.1

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

$m (s) = - 5 (s \cdot s^{4}) - 5 s^{4} \cdot 5 + 10 s^{3} s + 10 s^{3} \cdot 5 + 65 s^{2} s + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.1.2.2

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

$m (s) = - 5 s^{1 + 4} - 5 s^{4} \cdot 5 + 10 s^{3} s + 10 s^{3} \cdot 5 + 65 s^{2} s + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.1.3

Add $1$ and $4$ .

$m (s) = - 5 s^{5} - 5 s^{4} \cdot 5 + 10 s^{3} s + 10 s^{3} \cdot 5 + 65 s^{2} s + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.2

Multiply $5$ by $- 5$ .

$m (s) = - 5 s^{5} - 25 s^{4} + 10 s^{3} s + 10 s^{3} \cdot 5 + 65 s^{2} s + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.3

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

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Step 1.9.1.3.1

Move $s$ .

$m (s) = - 5 s^{5} - 25 s^{4} + 10 (s \cdot s^{3}) + 10 s^{3} \cdot 5 + 65 s^{2} s + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.3.2

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

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Step 1.9.1.3.2.1

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

$m (s) = - 5 s^{5} - 25 s^{4} + 10 (s \cdot s^{3}) + 10 s^{3} \cdot 5 + 65 s^{2} s + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.3.2.2

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

$m (s) = - 5 s^{5} - 25 s^{4} + 10 s^{1 + 3} + 10 s^{3} \cdot 5 + 65 s^{2} s + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.3.3

Add $1$ and $3$ .

$m (s) = - 5 s^{5} - 25 s^{4} + 10 s^{4} + 10 s^{3} \cdot 5 + 65 s^{2} s + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.4

Multiply $5$ by $10$ .

$m (s) = - 5 s^{5} - 25 s^{4} + 10 s^{4} + 50 s^{3} + 65 s^{2} s + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.5

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

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Step 1.9.1.5.1

Move $s$ .

$m (s) = - 5 s^{5} - 25 s^{4} + 10 s^{4} + 50 s^{3} + 65 (s \cdot s^{2}) + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.5.2

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

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Step 1.9.1.5.2.1

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

$m (s) = - 5 s^{5} - 25 s^{4} + 10 s^{4} + 50 s^{3} + 65 (s \cdot s^{2}) + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.5.2.2

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

$m (s) = - 5 s^{5} - 25 s^{4} + 10 s^{4} + 50 s^{3} + 65 s^{1 + 2} + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.5.3

Add $1$ and $2$ .

$m (s) = - 5 s^{5} - 25 s^{4} + 10 s^{4} + 50 s^{3} + 65 s^{3} + 65 s^{2} \cdot 5 - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.6

Multiply $5$ by $65$ .

$m (s) = - 5 s^{5} - 25 s^{4} + 10 s^{4} + 50 s^{3} + 65 s^{3} + 325 s^{2} - 70 s \cdot s - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.7

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

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Step 1.9.1.7.1

Move $s$ .

$m (s) = - 5 s^{5} - 25 s^{4} + 10 s^{4} + 50 s^{3} + 65 s^{3} + 325 s^{2} - 70 (s \cdot s) - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.7.2

Multiply $s$ by $s$ .

$m (s) = - 5 s^{5} - 25 s^{4} + 10 s^{4} + 50 s^{3} + 65 s^{3} + 325 s^{2} - 70 s^{2} - 70 s \cdot 5 - 120 s - 120 \cdot 5$

Step 1.9.1.8

Multiply $5$ by $- 70$ .

$m (s) = - 5 s^{5} - 25 s^{4} + 10 s^{4} + 50 s^{3} + 65 s^{3} + 325 s^{2} - 70 s^{2} - 350 s - 120 s - 120 \cdot 5$

Step 1.9.1.9

Multiply $- 120$ by $5$ .

$m (s) = - 5 s^{5} - 25 s^{4} + 10 s^{4} + 50 s^{3} + 65 s^{3} + 325 s^{2} - 70 s^{2} - 350 s - 120 s - 600$

Step 1.9.2

Simplify by adding terms.

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Step 1.9.2.1

Add $- 25 s^{4}$ and $10 s^{4}$ .

$m (s) = - 5 s^{5} - 15 s^{4} + 50 s^{3} + 65 s^{3} + 325 s^{2} - 70 s^{2} - 350 s - 120 s - 600$

Step 1.9.2.2

Add $50 s^{3}$ and $65 s^{3}$ .

$m (s) = - 5 s^{5} - 15 s^{4} + 115 s^{3} + 325 s^{2} - 70 s^{2} - 350 s - 120 s - 600$

Step 1.9.2.3

Subtract $70 s^{2}$ from $325 s^{2}$ .

$m (s) = - 5 s^{5} - 15 s^{4} + 115 s^{3} + 255 s^{2} - 350 s - 120 s - 600$

Step 1.9.2.4

Subtract $120 s$ from $- 350 s$ .

$m (s) = - 5 s^{5} - 15 s^{4} + 115 s^{3} + 255 s^{2} - 470 s - 600$

Step 2

The degree of a polynomial is the highest degree of its terms.

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Step 2.1

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$- 5 s^{5} \to 5$

$- 15 s^{4} \to 4$

$115 s^{3} \to 3$

$255 s^{2} \to 2$

$- 470 s \to 1$

$- 600 \to 0$

Step 2.2

The largest exponent is the degree of the polynomial.

$5$

Step 3

The leading term in a polynomial is the term with the highest degree.

$- 5 s^{5}$

Step 4

The leading coefficient of a polynomial is the coefficient of the leading term.

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Step 4.1

The leading term in a polynomial is the term with the highest degree.

$- 5 s^{5}$

Step 4.2

The leading coefficient in a polynomial is the coefficient of the leading term.

$- 5$

Step 5

List the results.

Polynomial Degree: $5$

Leading Term: $- 5 s^{5}$

Leading Coefficient: $- 5$