Basic Math Examples

$3 s^{3} + 16 s^{2} + 13 s - 19 \div s + 4$

Step 1

Rewrite the division as a fraction.

$3 s^{3} + 16 s^{2} + 13 s - \frac{19}{s} + 4$

Step 2

To write $3 s^{3}$ as a fraction with a common denominator, multiply by $\frac{s}{s}$ .

$16 s^{2} + 13 s + \frac{3 s^{3} s}{s} - \frac{19}{s} + 4$

Step 3

Combine the numerators over the common denominator.

$16 s^{2} + 13 s + \frac{3 s^{3} s - 19}{s} + 4$

Step 4

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

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Step 4.1

Move $s$ .

$16 s^{2} + 13 s + \frac{3 (s \cdot s^{3}) - 19}{s} + 4$

Step 4.2

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

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Step 4.2.1

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

$16 s^{2} + 13 s + \frac{3 (s^{1} s^{3}) - 19}{s} + 4$

Step 4.2.2

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

$16 s^{2} + 13 s + \frac{3 s^{1 + 3} - 19}{s} + 4$

Step 4.3

Add $1$ and $3$ .

$16 s^{2} + 13 s + \frac{3 s^{4} - 19}{s} + 4$

Step 5

To write $16 s^{2}$ as a fraction with a common denominator, multiply by $\frac{s}{s}$ .

$13 s + \frac{16 s^{2} s}{s} + \frac{3 s^{4} - 19}{s} + 4$

Step 6

Combine the numerators over the common denominator.

$13 s + \frac{16 s^{2} s + 3 s^{4} - 19}{s} + 4$

Step 7

Simplify the numerator.

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Step 7.1

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

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Step 7.1.1

Move $s$ .

$13 s + \frac{16 (s \cdot s^{2}) + 3 s^{4} - 19}{s} + 4$

Step 7.1.2

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

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Step 7.1.2.1

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

$13 s + \frac{16 (s^{1} s^{2}) + 3 s^{4} - 19}{s} + 4$

Step 7.1.2.2

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

$13 s + \frac{16 s^{1 + 2} + 3 s^{4} - 19}{s} + 4$

Step 7.1.3

Add $1$ and $2$ .

$13 s + \frac{16 s^{3} + 3 s^{4} - 19}{s} + 4$

Step 7.2

Reorder terms.

$13 s + \frac{3 s^{4} + 16 s^{3} - 19}{s} + 4$

Step 7.3

Factor $3 s^{4} + 16 s^{3} - 19$ using the rational roots test.

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Step 7.3.1

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.

$p = \pm 1, \pm 19$

$q = \pm 1, \pm 3$

Step 7.3.2

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

$\pm 1, \pm 0.\overline{3}, \pm 19, \pm 6.\overline{3}$

Step 7.3.3

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 7.3.3.1

Substitute $1$ into the polynomial.

$3 \cdot 1^{4} + 16 \cdot 1^{3} - 19$

Step 7.3.3.2

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

$3 \cdot 1 + 16 \cdot 1^{3} - 19$

Step 7.3.3.3

Multiply $3$ by $1$ .

$3 + 16 \cdot 1^{3} - 19$

Step 7.3.3.4

Raise $1$ to the power of $3$ .

$3 + 16 \cdot 1 - 19$

Step 7.3.3.5

Multiply $16$ by $1$ .

$3 + 16 - 19$

Step 7.3.3.6

Add $3$ and $16$ .

$19 - 19$

Step 7.3.3.7

Subtract $19$ from $19$ .

$0$

Step 7.3.4

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.

$\frac{3 s^{4} + 16 s^{3} - 19}{s - 1}$

Step 7.3.5

Divide $3 s^{4} + 16 s^{3} - 19$ by $s - 1$ .

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Step 7.3.5.1

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

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

Step 7.3.5.2

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

$3 s^{3}$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

Step 7.3.5.3

Multiply the new quotient term by the divisor.

$3 s^{3}$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

Step 7.3.5.4

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

$3 s^{3}$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

Step 7.3.5.5

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

$3 s^{3}$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

Step 7.3.5.6

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

$3 s^{3}$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

Step 7.3.5.7

Divide the highest order term in the dividend $19 s^{3}$ by the highest order term in divisor $s$ .

$3 s^{3}$

$19 s^{2}$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

Step 7.3.5.8

Multiply the new quotient term by the divisor.

$3 s^{3}$

$19 s^{2}$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

$19 s^{3}$

$19 s^{2}$

Step 7.3.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $19 s^{3} - 19 s^{2}$

$3 s^{3}$

$19 s^{2}$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

$19 s^{3}$

$19 s^{2}$

Step 7.3.5.10

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

$3 s^{3}$

$19 s^{2}$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

$19 s^{3}$

$19 s^{2}$

Step 7.3.5.11

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

$3 s^{3}$

$19 s^{2}$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

$19 s^{3}$

$19 s^{2}$

$0 s$

Step 7.3.5.12

Divide the highest order term in the dividend $19 s^{2}$ by the highest order term in divisor $s$ .

$3 s^{3}$

$19 s^{2}$

$19 s$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

$19 s^{3}$

$19 s^{2}$

$0 s$

Step 7.3.5.13

Multiply the new quotient term by the divisor.

$3 s^{3}$

$19 s^{2}$

$19 s$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

$19 s^{3}$

$19 s^{2}$

$0 s$

$19 s^{2}$

$19 s$

Step 7.3.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $19 s^{2} - 19 s$

$3 s^{3}$

$19 s^{2}$

$19 s$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

$19 s^{3}$

$19 s^{2}$

$0 s$

$19 s^{2}$

$19 s$

Step 7.3.5.15

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

$3 s^{3}$

$19 s^{2}$

$19 s$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

$19 s^{3}$

$19 s^{2}$

$0 s$

$19 s^{2}$

$19 s$

Step 7.3.5.16

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

$3 s^{3}$

$19 s^{2}$

$19 s$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

$19 s^{3}$

$19 s^{2}$

$0 s$

$19 s^{2}$

$19 s$

$19$

Step 7.3.5.17

Divide the highest order term in the dividend $19 s$ by the highest order term in divisor $s$ .

$3 s^{3}$

$19 s^{2}$

$19 s$

$19$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

$19 s^{3}$

$19 s^{2}$

$0 s$

$19 s^{2}$

$19 s$

$19$

Step 7.3.5.18

Multiply the new quotient term by the divisor.

$3 s^{3}$

$19 s^{2}$

$19 s$

$19$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

$19 s^{3}$

$19 s^{2}$

$0 s$

$19 s^{2}$

$19 s$

$19$

$19 s$

$19$

Step 7.3.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $19 s - 19$

$3 s^{3}$

$19 s^{2}$

$19 s$

$19$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

$19 s^{3}$

$19 s^{2}$

$0 s$

$19 s^{2}$

$19 s$

$19$

$19 s$

$19$

Step 7.3.5.20

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

$3 s^{3}$

$19 s^{2}$

$19 s$

$19$

$s$

$1$

$3 s^{4}$

$16 s^{3}$

$0 s^{2}$

$0 s$

$19$

$3 s^{4}$

$3 s^{3}$

$19 s^{3}$

$0 s^{2}$

$19 s^{3}$

$19 s^{2}$

$0 s$

$19 s^{2}$

$19 s$

$19$

$19 s$

$19$

$0$

Step 7.3.5.21

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

$3 s^{3} + 19 s^{2} + 19 s + 19$

Step 7.3.6

Write $3 s^{4} + 16 s^{3} - 19$ as a set of factors.

$13 s + \frac{(s - 1) (3 s^{3} + 19 s^{2} + 19 s + 19)}{s} + 4$

Step 8

Find the common denominator.

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Step 8.1

Write $13 s$ as a fraction with denominator $1$ .

$\frac{13 s}{1} + \frac{(s - 1) (3 s^{3} + 19 s^{2} + 19 s + 19)}{s} + 4$

Step 8.2

Multiply $\frac{13 s}{1}$ by $\frac{s}{s}$ .

$\frac{13 s}{1} \cdot \frac{s}{s} + \frac{(s - 1) (3 s^{3} + 19 s^{2} + 19 s + 19)}{s} + 4$

Step 8.3

Multiply $\frac{13 s}{1}$ by $\frac{s}{s}$ .

$\frac{13 s \cdot s}{s} + \frac{(s - 1) (3 s^{3} + 19 s^{2} + 19 s + 19)}{s} + 4$

Step 8.4

Write $4$ as a fraction with denominator $1$ .

$\frac{13 s \cdot s}{s} + \frac{(s - 1) (3 s^{3} + 19 s^{2} + 19 s + 19)}{s} + \frac{4}{1}$

Step 8.5

Multiply $\frac{4}{1}$ by $\frac{s}{s}$ .

$\frac{13 s \cdot s}{s} + \frac{(s - 1) (3 s^{3} + 19 s^{2} + 19 s + 19)}{s} + \frac{4}{1} \cdot \frac{s}{s}$

Step 8.6

Multiply $\frac{4}{1}$ by $\frac{s}{s}$ .

$\frac{13 s \cdot s}{s} + \frac{(s - 1) (3 s^{3} + 19 s^{2} + 19 s + 19)}{s} + \frac{4 s}{s}$

Step 9

Combine the numerators over the common denominator.

$\frac{13 s \cdot s + (s - 1) (3 s^{3} + 19 s^{2} + 19 s + 19) + 4 s}{s}$

Step 10

Simplify each term.

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Step 10.1

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

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Step 10.1.1

Move $s$ .

$\frac{13 (s \cdot s) + (s - 1) (3 s^{3} + 19 s^{2} + 19 s + 19) + 4 s}{s}$

Step 10.1.2

Multiply $s$ by $s$ .

$\frac{13 s^{2} + (s - 1) (3 s^{3} + 19 s^{2} + 19 s + 19) + 4 s}{s}$

Step 10.2

Expand $(s - 1) (3 s^{3} + 19 s^{2} + 19 s + 19)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{13 s^{2} + s (3 s^{3}) + s (19 s^{2}) + s (19 s) + s \cdot 19 - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3

Simplify each term.

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Step 10.3.1

Rewrite using the commutative property of multiplication.

$\frac{13 s^{2} + 3 s \cdot s^{3} + s (19 s^{2}) + s (19 s) + s \cdot 19 - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.2

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

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Step 10.3.2.1

Move $s^{3}$ .

$\frac{13 s^{2} + 3 (s^{3} s) + s (19 s^{2}) + s (19 s) + s \cdot 19 - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.2.2

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

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Step 10.3.2.2.1

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

$\frac{13 s^{2} + 3 (s^{3} s^{1}) + s (19 s^{2}) + s (19 s) + s \cdot 19 - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.2.2.2

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

$\frac{13 s^{2} + 3 s^{3 + 1} + s (19 s^{2}) + s (19 s) + s \cdot 19 - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.2.3

Add $3$ and $1$ .

$\frac{13 s^{2} + 3 s^{4} + s (19 s^{2}) + s (19 s) + s \cdot 19 - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.3

Rewrite using the commutative property of multiplication.

$\frac{13 s^{2} + 3 s^{4} + 19 s \cdot s^{2} + s (19 s) + s \cdot 19 - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.4

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

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Step 10.3.4.1

Move $s^{2}$ .

$\frac{13 s^{2} + 3 s^{4} + 19 (s^{2} s) + s (19 s) + s \cdot 19 - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.4.2

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

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Step 10.3.4.2.1

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

$\frac{13 s^{2} + 3 s^{4} + 19 (s^{2} s^{1}) + s (19 s) + s \cdot 19 - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.4.2.2

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

$\frac{13 s^{2} + 3 s^{4} + 19 s^{2 + 1} + s (19 s) + s \cdot 19 - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.4.3

Add $2$ and $1$ .

$\frac{13 s^{2} + 3 s^{4} + 19 s^{3} + s (19 s) + s \cdot 19 - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.5

Rewrite using the commutative property of multiplication.

$\frac{13 s^{2} + 3 s^{4} + 19 s^{3} + 19 s \cdot s + s \cdot 19 - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.6

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

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Step 10.3.6.1

Move $s$ .

$\frac{13 s^{2} + 3 s^{4} + 19 s^{3} + 19 (s \cdot s) + s \cdot 19 - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.6.2

Multiply $s$ by $s$ .

$\frac{13 s^{2} + 3 s^{4} + 19 s^{3} + 19 s^{2} + s \cdot 19 - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.7

Move $19$ to the left of $s$ .

$\frac{13 s^{2} + 3 s^{4} + 19 s^{3} + 19 s^{2} + 19 \cdot s - 1 (3 s^{3}) - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.8

Multiply $3$ by $- 1$ .

$\frac{13 s^{2} + 3 s^{4} + 19 s^{3} + 19 s^{2} + 19 s - 3 s^{3} - 1 (19 s^{2}) - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.9

Multiply $19$ by $- 1$ .

$\frac{13 s^{2} + 3 s^{4} + 19 s^{3} + 19 s^{2} + 19 s - 3 s^{3} - 19 s^{2} - 1 (19 s) - 1 \cdot 19 + 4 s}{s}$

Step 10.3.10

Multiply $19$ by $- 1$ .

$\frac{13 s^{2} + 3 s^{4} + 19 s^{3} + 19 s^{2} + 19 s - 3 s^{3} - 19 s^{2} - 19 s - 1 \cdot 19 + 4 s}{s}$

Step 10.3.11

Multiply $- 1$ by $19$ .

$\frac{13 s^{2} + 3 s^{4} + 19 s^{3} + 19 s^{2} + 19 s - 3 s^{3} - 19 s^{2} - 19 s - 19 + 4 s}{s}$

Step 10.4

Combine the opposite terms in $3 s^{4} + 19 s^{3} + 19 s^{2} + 19 s - 3 s^{3} - 19 s^{2} - 19 s - 19$ .

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Step 10.4.1

Subtract $19 s^{2}$ from $19 s^{2}$ .

$\frac{13 s^{2} + 3 s^{4} + 19 s^{3} + 19 s - 3 s^{3} + 0 - 19 s - 19 + 4 s}{s}$

Step 10.4.2

Add $3 s^{4} + 19 s^{3} + 19 s - 3 s^{3}$ and $0$ .

$\frac{13 s^{2} + 3 s^{4} + 19 s^{3} + 19 s - 3 s^{3} - 19 s - 19 + 4 s}{s}$

Step 10.4.3

Subtract $19 s$ from $19 s$ .

$\frac{13 s^{2} + 3 s^{4} + 19 s^{3} - 3 s^{3} + 0 - 19 + 4 s}{s}$

Step 10.4.4

Add $3 s^{4} + 19 s^{3} - 3 s^{3}$ and $0$ .

$\frac{13 s^{2} + 3 s^{4} + 19 s^{3} - 3 s^{3} - 19 + 4 s}{s}$

Step 10.5

Subtract $3 s^{3}$ from $19 s^{3}$ .

$\frac{13 s^{2} + 3 s^{4} + 16 s^{3} - 19 + 4 s}{s}$

Step 11

Reorder terms.

$\frac{3 s^{4} + 16 s^{3} + 13 s^{2} + 4 s - 19}{s}$