Calculus Examples

$f (x) = x^{2} {(x + 3)}^{3}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = {(x + 3)}^{3}$ .

$x^{2} \frac{d}{d x} [{(x + 3)}^{3}] + {(x + 3)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x + 3$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x + 3$ .

$x^{2} (\frac{d}{d u} [u^{3}] \frac{d}{d x} [x + 3]) + {(x + 3)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$x^{2} (3 u^{2} \frac{d}{d x} [x + 3]) + {(x + 3)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.2.3

Replace all occurrences of $u$ with $x + 3$ .

$x^{2} (3 {(x + 3)}^{2} \frac{d}{d x} [x + 3]) + {(x + 3)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$x^{2} (3 {(x + 3)}^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [3])) + {(x + 3)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x^{2} (3 {(x + 3)}^{2} (1 + \frac{d}{d x} [3])) + {(x + 3)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.3.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$x^{2} (3 {(x + 3)}^{2} (1 + 0)) + {(x + 3)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.3.4

Simplify the expression.

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Step 1.1.3.4.1

Add $1$ and $0$ .

$x^{2} (3 {(x + 3)}^{2} \cdot 1) + {(x + 3)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.3.4.2

Multiply $3$ by $1$ .

$x^{2} (3 {(x + 3)}^{2}) + {(x + 3)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{2} (3 {(x + 3)}^{2}) + {(x + 3)}^{3} (2 x)$

Step 1.1.3.6

Move $2$ to the left of ${(x + 3)}^{3}$ .

$x^{2} (3 {(x + 3)}^{2}) + 2 \cdot {(x + 3)}^{3} x$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Factor $x {(x + 3)}^{2}$ out of $x^{2} (3 {(x + 3)}^{2}) + 2 {(x + 3)}^{3} x$ .

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Step 1.1.4.1.1

Factor $x {(x + 3)}^{2}$ out of $x^{2} (3 {(x + 3)}^{2})$ .

$x {(x + 3)}^{2} (x \cdot (3)) + 2 {(x + 3)}^{3} x$

Step 1.1.4.1.2

Factor $x {(x + 3)}^{2}$ out of $2 {(x + 3)}^{3} x$ .

$x {(x + 3)}^{2} (x \cdot (3)) + x {(x + 3)}^{2} (2 (x + 3))$

Step 1.1.4.1.3

Factor $x {(x + 3)}^{2}$ out of $x {(x + 3)}^{2} (x \cdot (3)) + x {(x + 3)}^{2} (2 (x + 3))$ .

$x {(x + 3)}^{2} (x \cdot (3) + 2 (x + 3))$

Step 1.1.4.2

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

$x {(x + 3)}^{2} (3 x + 2 (x + 3))$

Step 1.1.4.3

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$x ((x + 3) (x + 3)) (3 x + 2 (x + 3))$

Step 1.1.4.4

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 1.1.4.4.1

Apply the distributive property.

$x (x (x + 3) + 3 (x + 3)) (3 x + 2 (x + 3))$

Step 1.1.4.4.2

Apply the distributive property.

$x (x \cdot x + x \cdot 3 + 3 (x + 3)) (3 x + 2 (x + 3))$

Step 1.1.4.4.3

Apply the distributive property.

$x (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) (3 x + 2 (x + 3))$

Step 1.1.4.5

Simplify and combine like terms.

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Step 1.1.4.5.1

Simplify each term.

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Step 1.1.4.5.1.1

Multiply $x$ by $x$ .

$x (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) (3 x + 2 (x + 3))$

Step 1.1.4.5.1.2

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

$x (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) (3 x + 2 (x + 3))$

Step 1.1.4.5.1.3

Multiply $3$ by $3$ .

$x (x^{2} + 3 x + 3 x + 9) (3 x + 2 (x + 3))$

Step 1.1.4.5.2

Add $3 x$ and $3 x$ .

$x (x^{2} + 6 x + 9) (3 x + 2 (x + 3))$

Step 1.1.4.6

Apply the distributive property.

$(x \cdot x^{2} + x (6 x) + x \cdot 9) (3 x + 2 (x + 3))$

Step 1.1.4.7

Simplify.

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Step 1.1.4.7.1

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

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Step 1.1.4.7.1.1

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

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Step 1.1.4.7.1.1.1

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

$(x^{1} x^{2} + x (6 x) + x \cdot 9) (3 x + 2 (x + 3))$

Step 1.1.4.7.1.1.2

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

$(x^{1 + 2} + x (6 x) + x \cdot 9) (3 x + 2 (x + 3))$

Step 1.1.4.7.1.2

Add $1$ and $2$ .

$(x^{3} + x (6 x) + x \cdot 9) (3 x + 2 (x + 3))$

Step 1.1.4.7.2

Rewrite using the commutative property of multiplication.

$(x^{3} + 6 x \cdot x + x \cdot 9) (3 x + 2 (x + 3))$

Step 1.1.4.7.3

Move $9$ to the left of $x$ .

$(x^{3} + 6 x \cdot x + 9 \cdot x) (3 x + 2 (x + 3))$

Step 1.1.4.8

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

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Step 1.1.4.8.1

Move $x$ .

$(x^{3} + 6 (x \cdot x) + 9 \cdot x) (3 x + 2 (x + 3))$

Step 1.1.4.8.2

Multiply $x$ by $x$ .

$(x^{3} + 6 x^{2} + 9 \cdot x) (3 x + 2 (x + 3))$

$(x^{3} + 6 x^{2} + 9 x) (3 x + 2 (x + 3))$

Step 1.1.4.9

Simplify each term.

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Step 1.1.4.9.1

Apply the distributive property.

$(x^{3} + 6 x^{2} + 9 x) (3 x + 2 x + 2 \cdot 3)$

Step 1.1.4.9.2

Multiply $2$ by $3$ .

$(x^{3} + 6 x^{2} + 9 x) (3 x + 2 x + 6)$

Step 1.1.4.10

Add $3 x$ and $2 x$ .

$(x^{3} + 6 x^{2} + 9 x) (5 x + 6)$

Step 1.1.4.11

Expand $(x^{3} + 6 x^{2} + 9 x) (5 x + 6)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} (5 x) + x^{3} \cdot 6 + 6 x^{2} (5 x) + 6 x^{2} \cdot 6 + 9 x (5 x) + 9 x \cdot 6$

Step 1.1.4.12

Simplify each term.

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Step 1.1.4.12.1

Rewrite using the commutative property of multiplication.

$5 x^{3} x + x^{3} \cdot 6 + 6 x^{2} (5 x) + 6 x^{2} \cdot 6 + 9 x (5 x) + 9 x \cdot 6$

Step 1.1.4.12.2

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

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Step 1.1.4.12.2.1

Move $x$ .

$5 (x \cdot x^{3}) + x^{3} \cdot 6 + 6 x^{2} (5 x) + 6 x^{2} \cdot 6 + 9 x (5 x) + 9 x \cdot 6$

Step 1.1.4.12.2.2

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

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Step 1.1.4.12.2.2.1

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

$5 (x^{1} x^{3}) + x^{3} \cdot 6 + 6 x^{2} (5 x) + 6 x^{2} \cdot 6 + 9 x (5 x) + 9 x \cdot 6$

Step 1.1.4.12.2.2.2

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

$5 x^{1 + 3} + x^{3} \cdot 6 + 6 x^{2} (5 x) + 6 x^{2} \cdot 6 + 9 x (5 x) + 9 x \cdot 6$

Step 1.1.4.12.2.3

Add $1$ and $3$ .

$5 x^{4} + x^{3} \cdot 6 + 6 x^{2} (5 x) + 6 x^{2} \cdot 6 + 9 x (5 x) + 9 x \cdot 6$

Step 1.1.4.12.3

Move $6$ to the left of $x^{3}$ .

$5 x^{4} + 6 \cdot x^{3} + 6 x^{2} (5 x) + 6 x^{2} \cdot 6 + 9 x (5 x) + 9 x \cdot 6$

Step 1.1.4.12.4

Rewrite using the commutative property of multiplication.

$5 x^{4} + 6 x^{3} + 6 \cdot 5 x^{2} x + 6 x^{2} \cdot 6 + 9 x (5 x) + 9 x \cdot 6$

Step 1.1.4.12.5

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

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Step 1.1.4.12.5.1

Move $x$ .

$5 x^{4} + 6 x^{3} + 6 \cdot 5 (x \cdot x^{2}) + 6 x^{2} \cdot 6 + 9 x (5 x) + 9 x \cdot 6$

Step 1.1.4.12.5.2

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

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Step 1.1.4.12.5.2.1

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

$5 x^{4} + 6 x^{3} + 6 \cdot 5 (x^{1} x^{2}) + 6 x^{2} \cdot 6 + 9 x (5 x) + 9 x \cdot 6$

Step 1.1.4.12.5.2.2

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

$5 x^{4} + 6 x^{3} + 6 \cdot 5 x^{1 + 2} + 6 x^{2} \cdot 6 + 9 x (5 x) + 9 x \cdot 6$

Step 1.1.4.12.5.3

Add $1$ and $2$ .

$5 x^{4} + 6 x^{3} + 6 \cdot 5 x^{3} + 6 x^{2} \cdot 6 + 9 x (5 x) + 9 x \cdot 6$

Step 1.1.4.12.6

Multiply $6$ by $5$ .

$5 x^{4} + 6 x^{3} + 30 x^{3} + 6 x^{2} \cdot 6 + 9 x (5 x) + 9 x \cdot 6$

Step 1.1.4.12.7

Multiply $6$ by $6$ .

$5 x^{4} + 6 x^{3} + 30 x^{3} + 36 x^{2} + 9 x (5 x) + 9 x \cdot 6$

Step 1.1.4.12.8

Rewrite using the commutative property of multiplication.

$5 x^{4} + 6 x^{3} + 30 x^{3} + 36 x^{2} + 9 \cdot 5 x \cdot x + 9 x \cdot 6$

Step 1.1.4.12.9

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

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Step 1.1.4.12.9.1

Move $x$ .

$5 x^{4} + 6 x^{3} + 30 x^{3} + 36 x^{2} + 9 \cdot 5 (x \cdot x) + 9 x \cdot 6$

Step 1.1.4.12.9.2

Multiply $x$ by $x$ .

$5 x^{4} + 6 x^{3} + 30 x^{3} + 36 x^{2} + 9 \cdot 5 x^{2} + 9 x \cdot 6$

Step 1.1.4.12.10

Multiply $9$ by $5$ .

$5 x^{4} + 6 x^{3} + 30 x^{3} + 36 x^{2} + 45 x^{2} + 9 x \cdot 6$

Step 1.1.4.12.11

Multiply $6$ by $9$ .

$5 x^{4} + 6 x^{3} + 30 x^{3} + 36 x^{2} + 45 x^{2} + 54 x$

Step 1.1.4.13

Add $6 x^{3}$ and $30 x^{3}$ .

$5 x^{4} + 36 x^{3} + 36 x^{2} + 45 x^{2} + 54 x$

Step 1.1.4.14

Add $36 x^{2}$ and $45 x^{2}$ .

$f' (x) = 5 x^{4} + 36 x^{3} + 81 x^{2} + 54 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $5 x^{4} + 36 x^{3} + 81 x^{2} + 54 x$ .

$5 x^{4} + 36 x^{3} + 81 x^{2} + 54 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $5 x^{4} + 36 x^{3} + 81 x^{2} + 54 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$5 x^{4} + 36 x^{3} + 81 x^{2} + 54 x = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $x$ out of $5 x^{4} + 36 x^{3} + 81 x^{2} + 54 x$ .

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Step 2.2.1.1

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

$x (5 x^{3}) + 36 x^{3} + 81 x^{2} + 54 x = 0$

Step 2.2.1.2

Factor $x$ out of $36 x^{3}$ .

$x (5 x^{3}) + x (36 x^{2}) + 81 x^{2} + 54 x = 0$

Step 2.2.1.3

Factor $x$ out of $81 x^{2}$ .

$x (5 x^{3}) + x (36 x^{2}) + x (81 x) + 54 x = 0$

Step 2.2.1.4

Factor $x$ out of $54 x$ .

$x (5 x^{3}) + x (36 x^{2}) + x (81 x) + x \cdot 54 = 0$

Step 2.2.1.5

Factor $x$ out of $x (5 x^{3}) + x (36 x^{2})$ .

$x (5 x^{3} + 36 x^{2}) + x (81 x) + x \cdot 54 = 0$

Step 2.2.1.6

Factor $x$ out of $x (5 x^{3} + 36 x^{2}) + x (81 x)$ .

$x (5 x^{3} + 36 x^{2} + 81 x) + x \cdot 54 = 0$

Step 2.2.1.7

Factor $x$ out of $x (5 x^{3} + 36 x^{2} + 81 x) + x \cdot 54$ .

$x (5 x^{3} + 36 x^{2} + 81 x + 54) = 0$

Step 2.2.2

Factor $5 x^{3} + 36 x^{2} + 81 x + 54$ 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.

$p = \pm 1, \pm 54, \pm 2, \pm 27, \pm 3, \pm 18, \pm 6, \pm 9$

$q = \pm 1, \pm 5$

Step 2.2.2.2

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

$\pm 1, \pm 0.2, \pm 54, \pm 10.8, \pm 2, \pm 0.4, \pm 27, \pm 5.4, \pm 3, \pm 0.6, \pm 18, \pm 3.6, \pm 6, \pm 1.2, \pm 9, \pm 1.8$

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.

$5 {(- 3)}^{3} + 36 {(- 3)}^{2} + 81 \cdot - 3 + 54$

Step 2.2.2.3.2

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

$5 \cdot - 27 + 36 {(- 3)}^{2} + 81 \cdot - 3 + 54$

Step 2.2.2.3.3

Multiply $5$ by $- 27$ .

$- 135 + 36 {(- 3)}^{2} + 81 \cdot - 3 + 54$

Step 2.2.2.3.4

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

$- 135 + 36 \cdot 9 + 81 \cdot - 3 + 54$

Step 2.2.2.3.5

Multiply $36$ by $9$ .

$- 135 + 324 + 81 \cdot - 3 + 54$

Step 2.2.2.3.6

Add $- 135$ and $324$ .

$189 + 81 \cdot - 3 + 54$

Step 2.2.2.3.7

Multiply $81$ by $- 3$ .

$189 - 243 + 54$

Step 2.2.2.3.8

Subtract $243$ from $189$ .

$- 54 + 54$

Step 2.2.2.3.9

Add $- 54$ and $54$ .

$0$

Step 2.2.2.4

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.

$\frac{5 x^{3} + 36 x^{2} + 81 x + 54}{x + 3}$

Step 2.2.2.5

Divide $5 x^{3} + 36 x^{2} + 81 x + 54$ by $x + 3$ .

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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 $0$ .

$x$

$3$

$5 x^{3}$

$36 x^{2}$

$81 x$

$54$

Step 2.2.2.5.2

Divide the highest order term in the dividend $5 x^{3}$ by the highest order term in divisor $x$ .

					$5 x^{2}$
	$x$	+	$3$		$5 x^{3}$	+	$36 x^{2}$	+	$81 x$	+	$54$

Step 2.2.2.5.3

Multiply the new quotient term by the divisor.

				$5 x^{2}$
$x$	+	$3$		$5 x^{3}$	+	$36 x^{2}$	+	$81 x$	+	$54$
			+	$5 x^{3}$	+	$15 x^{2}$

Step 2.2.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} + 15 x^{2}$

				$5 x^{2}$
$x$	+	$3$		$5 x^{3}$	+	$36 x^{2}$	+	$81 x$	+	$54$
			-	$5 x^{3}$	-	$15 x^{2}$

Step 2.2.2.5.5

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

				$5 x^{2}$
$x$	+	$3$		$5 x^{3}$	+	$36 x^{2}$	+	$81 x$	+	$54$
			-	$5 x^{3}$	-	$15 x^{2}$
					+	$21 x^{2}$

Step 2.2.2.5.6

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

				$5 x^{2}$
$x$	+	$3$		$5 x^{3}$	+	$36 x^{2}$	+	$81 x$	+	$54$
			-	$5 x^{3}$	-	$15 x^{2}$
					+	$21 x^{2}$	+	$81 x$

Step 2.2.2.5.7

Divide the highest order term in the dividend $21 x^{2}$ by the highest order term in divisor $x$ .

				$5 x^{2}$	+	$21 x$
$x$	+	$3$		$5 x^{3}$	+	$36 x^{2}$	+	$81 x$	+	$54$
			-	$5 x^{3}$	-	$15 x^{2}$
					+	$21 x^{2}$	+	$81 x$

Step 2.2.2.5.8

Multiply the new quotient term by the divisor.

				$5 x^{2}$	+	$21 x$
$x$	+	$3$		$5 x^{3}$	+	$36 x^{2}$	+	$81 x$	+	$54$
			-	$5 x^{3}$	-	$15 x^{2}$
					+	$21 x^{2}$	+	$81 x$
					+	$21 x^{2}$	+	$63 x$

Step 2.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $21 x^{2} + 63 x$

				$5 x^{2}$	+	$21 x$
$x$	+	$3$		$5 x^{3}$	+	$36 x^{2}$	+	$81 x$	+	$54$
			-	$5 x^{3}$	-	$15 x^{2}$
					+	$21 x^{2}$	+	$81 x$
					-	$21 x^{2}$	-	$63 x$

Step 2.2.2.5.10

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

				$5 x^{2}$	+	$21 x$
$x$	+	$3$		$5 x^{3}$	+	$36 x^{2}$	+	$81 x$	+	$54$
			-	$5 x^{3}$	-	$15 x^{2}$
					+	$21 x^{2}$	+	$81 x$
					-	$21 x^{2}$	-	$63 x$
							+	$18 x$

Step 2.2.2.5.11

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

				$5 x^{2}$	+	$21 x$
$x$	+	$3$		$5 x^{3}$	+	$36 x^{2}$	+	$81 x$	+	$54$
			-	$5 x^{3}$	-	$15 x^{2}$
					+	$21 x^{2}$	+	$81 x$
					-	$21 x^{2}$	-	$63 x$
							+	$18 x$	+	$54$

Step 2.2.2.5.12

Divide the highest order term in the dividend $18 x$ by the highest order term in divisor $x$ .

				$5 x^{2}$	+	$21 x$	+	$18$
$x$	+	$3$		$5 x^{3}$	+	$36 x^{2}$	+	$81 x$	+	$54$
			-	$5 x^{3}$	-	$15 x^{2}$
					+	$21 x^{2}$	+	$81 x$
					-	$21 x^{2}$	-	$63 x$
							+	$18 x$	+	$54$

Step 2.2.2.5.13

Multiply the new quotient term by the divisor.

				$5 x^{2}$	+	$21 x$	+	$18$
$x$	+	$3$		$5 x^{3}$	+	$36 x^{2}$	+	$81 x$	+	$54$
			-	$5 x^{3}$	-	$15 x^{2}$
					+	$21 x^{2}$	+	$81 x$
					-	$21 x^{2}$	-	$63 x$
							+	$18 x$	+	$54$
							+	$18 x$	+	$54$

Step 2.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $18 x + 54$

				$5 x^{2}$	+	$21 x$	+	$18$
$x$	+	$3$		$5 x^{3}$	+	$36 x^{2}$	+	$81 x$	+	$54$
			-	$5 x^{3}$	-	$15 x^{2}$
					+	$21 x^{2}$	+	$81 x$
					-	$21 x^{2}$	-	$63 x$
							+	$18 x$	+	$54$
							-	$18 x$	-	$54$

Step 2.2.2.5.15

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

				$5 x^{2}$	+	$21 x$	+	$18$
$x$	+	$3$		$5 x^{3}$	+	$36 x^{2}$	+	$81 x$	+	$54$
			-	$5 x^{3}$	-	$15 x^{2}$
					+	$21 x^{2}$	+	$81 x$
					-	$21 x^{2}$	-	$63 x$
							+	$18 x$	+	$54$
							-	$18 x$	-	$54$
										$0$

Step 2.2.2.5.16

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

$5 x^{2} + 21 x + 18$

Step 2.2.2.6

Write $5 x^{3} + 36 x^{2} + 81 x + 54$ as a set of factors.

$x ((x + 3) (5 x^{2} + 21 x + 18)) = 0$

Step 2.2.3

Factor.

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Step 2.2.3.1

Factor by grouping.

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Step 2.2.3.1.1

Factor by grouping.

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Step 2.2.3.1.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot 18 = 90$ and whose sum is $b = 21$ .

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Step 2.2.3.1.1.1.1

Factor $21$ out of $21 x$ .

$x ((x + 3) (5 x^{2} + 21 (x) + 18)) = 0$

Step 2.2.3.1.1.1.2

Rewrite $21$ as $6$ plus $15$

$x ((x + 3) (5 x^{2} + (6 + 15) x + 18)) = 0$

Step 2.2.3.1.1.1.3

Apply the distributive property.

$x ((x + 3) (5 x^{2} + 6 x + 15 x + 18)) = 0$

Step 2.2.3.1.1.2

Factor out the greatest common factor from each group.

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Step 2.2.3.1.1.2.1

Group the first two terms and the last two terms.

$x ((x + 3) ((5 x^{2} + 6 x) + 15 x + 18)) = 0$

Step 2.2.3.1.1.2.2

Factor out the greatest common factor (GCF) from each group.

$x ((x + 3) (x (5 x + 6) + 3 (5 x + 6))) = 0$

Step 2.2.3.1.1.3

Factor the polynomial by factoring out the greatest common factor, $5 x + 6$ .

$x ((x + 3) ((5 x + 6) (x + 3))) = 0$

Step 2.2.3.1.2

Remove unnecessary parentheses.

$x ((x + 3) (5 x + 6) (x + 3)) = 0$

Step 2.2.3.2

Remove unnecessary parentheses.

$x (x + 3) (5 x + 6) (x + 3) = 0$

Step 2.2.4

Combine exponents.

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Step 2.2.4.1

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

$x ((x + 3) (x + 3)) (5 x + 6) = 0$

Step 2.2.4.2

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

$x ((x + 3) (x + 3)) (5 x + 6) = 0$

Step 2.2.4.3

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

$x {(x + 3)}^{1 + 1} (5 x + 6) = 0$

Step 2.2.4.4

Add $1$ and $1$ .

$x {(x + 3)}^{2} (5 x + 6) = 0$

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 $0$ .

$x = 0$

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

$5 x + 6 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

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

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Step 2.5.1

Set ${(x + 3)}^{2}$ equal to $0$ .

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

Step 2.5.2

Solve ${(x + 3)}^{2} = 0$ for $x$ .

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Step 2.5.2.1

Set the $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 2.5.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.6

Set $5 x + 6$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $5 x + 6$ equal to $0$ .

$5 x + 6 = 0$

Step 2.6.2

Solve $5 x + 6 = 0$ for $x$ .

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Step 2.6.2.1

Subtract $6$ from both sides of the equation.

$5 x = - 6$

Step 2.6.2.2

Divide each term in $5 x = - 6$ by $5$ and simplify.

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Step 2.6.2.2.1

Divide each term in $5 x = - 6$ by $5$ .

$\frac{5 x}{5} = \frac{- 6}{5}$

Step 2.6.2.2.2

Simplify the left side.

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Step 2.6.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.6.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 6}{5}$

Step 2.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{5}$

Step 2.6.2.2.3

Simplify the right side.

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Step 2.6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{6}{5}$

Step 2.7

The final solution is all the values that make $x {(x + 3)}^{2} (5 x + 6) = 0$ true.

$x = 0, - 3, - \frac{6}{5}$

Step 3

The values which make the derivative equal to $0$ are $0, - 3, - \frac{6}{5}$ .

$0, - 3, - \frac{6}{5}$

Step 4

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - 3) \cup (- 3, - \frac{6}{5}) \cup (- \frac{6}{5}, 0) \cup (0, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 3)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 4$ in the expression.

$f' (- 4) = 5 {(- 4)}^{4} + 36 {(- 4)}^{3} + 81 {(- 4)}^{2} + 54 (- 4)$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f' (- 4) = 5 \cdot 256 + 36 {(- 4)}^{3} + 81 {(- 4)}^{2} + 54 (- 4)$

Step 5.2.1.2

Multiply $5$ by $256$ .

$f' (- 4) = 1280 + 36 {(- 4)}^{3} + 81 {(- 4)}^{2} + 54 (- 4)$

Step 5.2.1.3

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

$f' (- 4) = 1280 + 36 \cdot - 64 + 81 {(- 4)}^{2} + 54 (- 4)$

Step 5.2.1.4

Multiply $36$ by $- 64$ .

$f' (- 4) = 1280 - 2304 + 81 {(- 4)}^{2} + 54 (- 4)$

Step 5.2.1.5

Raise $- 4$ to the power of $2$ .

$f' (- 4) = 1280 - 2304 + 81 \cdot 16 + 54 (- 4)$

Step 5.2.1.6

Multiply $81$ by $16$ .

$f' (- 4) = 1280 - 2304 + 1296 + 54 (- 4)$

Step 5.2.1.7

Multiply $54$ by $- 4$ .

$f' (- 4) = 1280 - 2304 + 1296 - 216$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $2304$ from $1280$ .

$f' (- 4) = - 1024 + 1296 - 216$

Step 5.2.2.2

Add $- 1024$ and $1296$ .

$f' (- 4) = 272 - 216$

Step 5.2.2.3

Subtract $216$ from $272$ .

$f' (- 4) = 56$

Step 5.2.3

The final answer is $56$ .

$56$

Step 5.3

At $x = - 4$ the derivative is $56$ . Since this is positive, the function is increasing on $(- \infty, - 3)$ .

Increasing on $(- \infty, - 3)$ since $f' (x) > 0$

Step 6

Substitute a value from the interval $(- 3, - \frac{6}{5})$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 2.1$ in the expression.

$f' (- 2.1) = 5 {(- 2.1)}^{4} + 36 {(- 2.1)}^{3} + 81 {(- 2.1)}^{2} + 54 (- 2.1)$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 2.1$ to the power of $4$ .

$f' (- 2.1) = 5 \cdot 19.4481 + 36 {(- 2.1)}^{3} + 81 {(- 2.1)}^{2} + 54 (- 2.1)$

Step 6.2.1.2

Multiply $5$ by $19.4481$ .

$f' (- 2.1) = 97.2405 + 36 {(- 2.1)}^{3} + 81 {(- 2.1)}^{2} + 54 (- 2.1)$

Step 6.2.1.3

Raise $- 2.1$ to the power of $3$ .

$f' (- 2.1) = 97.2405 + 36 \cdot - 9.261 + 81 {(- 2.1)}^{2} + 54 (- 2.1)$

Step 6.2.1.4

Multiply $36$ by $- 9.261$ .

$f' (- 2.1) = 97.2405 - 333.396 + 81 {(- 2.1)}^{2} + 54 (- 2.1)$

Step 6.2.1.5

Raise $- 2.1$ to the power of $2$ .

$f' (- 2.1) = 97.2405 - 333.396 + 81 \cdot 4.41 + 54 (- 2.1)$

Step 6.2.1.6

Multiply $81$ by $4.41$ .

$f' (- 2.1) = 97.2405 - 333.396 + 357.21 + 54 (- 2.1)$

Step 6.2.1.7

Multiply $54$ by $- 2.1$ .

$f' (- 2.1) = 97.2405 - 333.396 + 357.21 - 113.4$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $333.396$ from $97.2405$ .

$f' (- 2.1) = - 236.1555 + 357.21 - 113.4$

Step 6.2.2.2

Add $- 236.1555$ and $357.21$ .

$f' (- 2.1) = 121.0545 - 113.4$

Step 6.2.2.3

Subtract $113.4$ from $121.0545$ .

$f' (- 2.1) = 7.6545$

Step 6.2.3

The final answer is $7.6545$ .

$7.6545$

Step 6.3

At $x = - 2.1$ the derivative is $7.6545$ . Since this is positive, the function is increasing on $(- 3, - \frac{6}{5})$ .

Increasing on $(- 3, - \frac{6}{5})$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(- 1.2, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 0.6$ in the expression.

$f' (- 0.6) = 5 {(- 0.6)}^{4} + 36 {(- 0.6)}^{3} + 81 {(- 0.6)}^{2} + 54 (- 0.6)$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $- 0.6$ to the power of $4$ .

$f' (- 0.6) = 5 \cdot 0.1296 + 36 {(- 0.6)}^{3} + 81 {(- 0.6)}^{2} + 54 (- 0.6)$

Step 7.2.1.2

Multiply $5$ by $0.1296$ .

$f' (- 0.6) = 0.648 + 36 {(- 0.6)}^{3} + 81 {(- 0.6)}^{2} + 54 (- 0.6)$

Step 7.2.1.3

Raise $- 0.6$ to the power of $3$ .

$f' (- 0.6) = 0.648 + 36 \cdot - 0.216 + 81 {(- 0.6)}^{2} + 54 (- 0.6)$

Step 7.2.1.4

Multiply $36$ by $- 0.216$ .

$f' (- 0.6) = 0.648 - 7.776 + 81 {(- 0.6)}^{2} + 54 (- 0.6)$

Step 7.2.1.5

Raise $- 0.6$ to the power of $2$ .

$f' (- 0.6) = 0.648 - 7.776 + 81 \cdot 0.36 + 54 (- 0.6)$

Step 7.2.1.6

Multiply $81$ by $0.36$ .

$f' (- 0.6) = 0.648 - 7.776 + 29.16 + 54 (- 0.6)$

Step 7.2.1.7

Multiply $54$ by $- 0.6$ .

$f' (- 0.6) = 0.648 - 7.776 + 29.16 - 32.4$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $7.776$ from $0.648$ .

$f' (- 0.6) = - 7.128 + 29.16 - 32.4$

Step 7.2.2.2

Add $- 7.128$ and $29.16$ .

$f' (- 0.6) = 22.032 - 32.4$

Step 7.2.2.3

Subtract $32.4$ from $22.032$ .

$f' (- 0.6) = - 10.368$

Step 7.2.3

The final answer is $- 10.368$ .

$- 10.368$

Step 7.3

At $x = - 0.6$ the derivative is $- 10.368$ . Since this is negative, the function is decreasing on $(- 1.2, 0)$ .

Decreasing on $(- \frac{6}{5}, 0)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(0, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = 5 {(1)}^{4} + 36 {(1)}^{3} + 81 {(1)}^{2} + 54 (1)$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

One to any power is one.

$f' (1) = 5 \cdot 1 + 36 {(1)}^{3} + 81 {(1)}^{2} + 54 (1)$

Step 8.2.1.2

Multiply $5$ by $1$ .

$f' (1) = 5 + 36 {(1)}^{3} + 81 {(1)}^{2} + 54 (1)$

Step 8.2.1.3

One to any power is one.

$f' (1) = 5 + 36 \cdot 1 + 81 {(1)}^{2} + 54 (1)$

Step 8.2.1.4

Multiply $36$ by $1$ .

$f' (1) = 5 + 36 + 81 {(1)}^{2} + 54 (1)$

Step 8.2.1.5

One to any power is one.

$f' (1) = 5 + 36 + 81 \cdot 1 + 54 (1)$

Step 8.2.1.6

Multiply $81$ by $1$ .

$f' (1) = 5 + 36 + 81 + 54 (1)$

Step 8.2.1.7

Multiply $54$ by $1$ .

$f' (1) = 5 + 36 + 81 + 54$

Step 8.2.2

Simplify by adding numbers.

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Step 8.2.2.1

Add $5$ and $36$ .

$f' (1) = 41 + 81 + 54$

Step 8.2.2.2

Add $41$ and $81$ .

$f' (1) = 122 + 54$

Step 8.2.2.3

Add $122$ and $54$ .

$f' (1) = 176$

Step 8.2.3

The final answer is $176$ .

$176$

Step 8.3

At $x = 1$ the derivative is $176$ . Since this is positive, the function is increasing on $(0, \infty)$ .

Increasing on $(0, \infty)$ since $f' (x) > 0$

Step 9

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, - 3), (- 3, - \frac{6}{5}), (0, \infty)$

Decreasing on: $(- \frac{6}{5}, 0)$

Step 10