Calculus Examples

$f (x) = 3 x {(16 - x)}^{3}$

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

Since $3$ is constant with respect to $x$ , the derivative of $3 x {(16 - x)}^{3}$ with respect to $x$ is $3 \frac{d}{d x} [x {(16 - x)}^{3}]$ .

$f' (x) = 3 \frac{d}{d x} (x {(16 - x)}^{3})$

Step 1.1.2

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 $f (x) = x$ and $g (x) = {(16 - x)}^{3}$ .

$f' (x) = 3 (x \frac{d}{d x} ({(16 - x)}^{3}) + {(16 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.3

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 $g (x) = 16 - x$ .

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Step 1.1.3.1

To apply the Chain Rule, set $u$ as $16 - x$ .

$f' (x) = 3 (x (\frac{d}{d u} (u^{3}) \frac{d}{d x} (16 - x)) + {(16 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.3.2

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

$f' (x) = 3 (x (3 u^{2} \frac{d}{d x} (16 - x)) + {(16 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.3.3

Replace all occurrences of $u$ with $16 - x$ .

$f' (x) = 3 (x (3 {(16 - x)}^{2} \frac{d}{d x} (16 - x)) + {(16 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4

Differentiate.

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Step 1.1.4.1

By the Sum Rule, the derivative of $16 - x$ with respect to $x$ is $\frac{d}{d x} [16] + \frac{d}{d x} [- x]$ .

$f' (x) = 3 (x (3 {(16 - x)}^{2} (\frac{d}{d x} (16) + \frac{d}{d x} (- x))) + {(16 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.2

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$f' (x) = 3 (x (3 {(16 - x)}^{2} (0 + \frac{d}{d x} (- x))) + {(16 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.3

Add $0$ and $\frac{d}{d x} [- x]$ .

$f' (x) = 3 (x (3 {(16 - x)}^{2} \frac{d}{d x} (- x)) + {(16 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$f' (x) = 3 (x (3 {(16 - x)}^{2} (- \frac{d}{d x} x)) + {(16 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.5

Multiply $- 1$ by $3$ .

$f' (x) = 3 (x (- 3 {(16 - x)}^{2} \frac{d}{d x} x) + {(16 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.6

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

$f' (x) = 3 (x (- 3 {(16 - x)}^{2} \cdot 1) + {(16 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.7

Multiply $- 3$ by $1$ .

$f' (x) = 3 (x (- 3 {(16 - x)}^{2}) + {(16 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.8

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

$f' (x) = 3 (x (- 3 {(16 - x)}^{2}) + {(16 - x)}^{3} \cdot 1)$

Step 1.1.4.9

Multiply ${(16 - x)}^{3}$ by $1$ .

$f' (x) = 3 (x (- 3 {(16 - x)}^{2}) + {(16 - x)}^{3})$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Apply the distributive property.

$f' (x) = 3 (x (- 3 {(16 - x)}^{2})) + 3 {(16 - x)}^{3}$

Step 1.1.5.2

Multiply $- 3$ by $3$ .

$f' (x) = - 9 (x ({(16 - x)}^{2})) + 3 {(16 - x)}^{3}$

Step 1.1.5.3

Factor $3 {(16 - x)}^{2}$ out of $- 9 x {(16 - x)}^{2} + 3 {(16 - x)}^{3}$ .

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Step 1.1.5.3.1

Factor $3 {(16 - x)}^{2}$ out of $- 9 x {(16 - x)}^{2}$ .

$f' (x) = 3 {(16 - x)}^{2} (- 3 x) + 3 {(16 - x)}^{3}$

Step 1.1.5.3.2

Factor $3 {(16 - x)}^{2}$ out of $3 {(16 - x)}^{3}$ .

$f' (x) = 3 {(16 - x)}^{2} (- 3 x) + 3 {(16 - x)}^{2} (16 - x)$

Step 1.1.5.3.3

Factor $3 {(16 - x)}^{2}$ out of $3 {(16 - x)}^{2} (- 3 x) + 3 {(16 - x)}^{2} (16 - x)$ .

$f' (x) = 3 {(16 - x)}^{2} (- 3 x + 16 - x)$

Step 1.1.5.4

Subtract $x$ from $- 3 x$ .

$f' (x) = 3 {(16 - x)}^{2} (- 4 x + 16)$

Step 1.1.5.5

Rewrite ${(16 - x)}^{2}$ as $(16 - x) (16 - x)$ .

$f' (x) = 3 ((16 - x) (16 - x)) (- 4 x + 16)$

Step 1.1.5.6

Expand $(16 - x) (16 - x)$ using the FOIL Method.

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Step 1.1.5.6.1

Apply the distributive property.

$f' (x) = 3 (16 (16 - x) - x (16 - x)) (- 4 x + 16)$

Step 1.1.5.6.2

Apply the distributive property.

$f' (x) = 3 (16 \cdot 16 + 16 (- x) - x (16 - x)) (- 4 x + 16)$

Step 1.1.5.6.3

Apply the distributive property.

$f' (x) = 3 (16 \cdot 16 + 16 (- x) - x \cdot 16 - x (- x)) (- 4 x + 16)$

Step 1.1.5.7

Simplify and combine like terms.

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Step 1.1.5.7.1

Simplify each term.

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Step 1.1.5.7.1.1

Multiply $16$ by $16$ .

$f' (x) = 3 (256 + 16 (- x) - x \cdot 16 - x (- x)) (- 4 x + 16)$

Step 1.1.5.7.1.2

Multiply $- 1$ by $16$ .

$f' (x) = 3 (256 - 16 x - x \cdot 16 - x (- x)) (- 4 x + 16)$

Step 1.1.5.7.1.3

Multiply $16$ by $- 1$ .

$f' (x) = 3 (256 - 16 x - 16 x - x (- x)) (- 4 x + 16)$

Step 1.1.5.7.1.4

Rewrite using the commutative property of multiplication.

$f' (x) = 3 (256 - 16 x - 16 x - 1 \cdot (- 1 x \cdot x)) (- 4 x + 16)$

Step 1.1.5.7.1.5

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

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Step 1.1.5.7.1.5.1

Move $x$ .

$f' (x) = 3 (256 - 16 x - 16 x - 1 \cdot (- 1 (x \cdot x))) (- 4 x + 16)$

Step 1.1.5.7.1.5.2

Multiply $x$ by $x$ .

$f' (x) = 3 (256 - 16 x - 16 x - 1 \cdot (- 1 x^{2})) (- 4 x + 16)$

Step 1.1.5.7.1.6

Multiply $- 1$ by $- 1$ .

$f' (x) = 3 (256 - 16 x - 16 x + 1 x^{2}) (- 4 x + 16)$

Step 1.1.5.7.1.7

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

$f' (x) = 3 (256 - 16 x - 16 x + x^{2}) (- 4 x + 16)$

Step 1.1.5.7.2

Subtract $16 x$ from $- 16 x$ .

$f' (x) = 3 (256 - 32 x + x^{2}) (- 4 x + 16)$

Step 1.1.5.8

Apply the distributive property.

$f' (x) = (3 \cdot 256 + 3 (- 32 x) + 3 x^{2}) (- 4 x + 16)$

Step 1.1.5.9

Simplify.

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Step 1.1.5.9.1

Multiply $3$ by $256$ .

$f' (x) = (768 + 3 (- 32 x) + 3 x^{2}) (- 4 x + 16)$

Step 1.1.5.9.2

Multiply $- 32$ by $3$ .

$f' (x) = (768 - 96 x + 3 x^{2}) (- 4 x + 16)$

Step 1.1.5.10

Expand $(768 - 96 x + 3 x^{2}) (- 4 x + 16)$ by multiplying each term in the first expression by each term in the second expression.

$f' (x) = 768 (- 4 x) + 768 \cdot 16 - 96 x (- 4 x) - 96 x \cdot 16 + 3 x^{2} (- 4 x) + 3 x^{2} \cdot 16$

Step 1.1.5.11

Simplify each term.

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Step 1.1.5.11.1

Multiply $- 4$ by $768$ .

$f' (x) = - 3072 x + 768 \cdot 16 - 96 x (- 4 x) - 96 x \cdot 16 + 3 x^{2} (- 4 x) + 3 x^{2} \cdot 16$

Step 1.1.5.11.2

Multiply $768$ by $16$ .

$f' (x) = - 3072 x + 12288 - 96 x (- 4 x) - 96 x \cdot 16 + 3 x^{2} (- 4 x) + 3 x^{2} \cdot 16$

Step 1.1.5.11.3

Rewrite using the commutative property of multiplication.

$f' (x) = - 3072 x + 12288 - 96 \cdot (- 4 x \cdot x) - 96 x \cdot 16 + 3 x^{2} (- 4 x) + 3 x^{2} \cdot 16$

Step 1.1.5.11.4

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

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Step 1.1.5.11.4.1

Move $x$ .

$f' (x) = - 3072 x + 12288 - 96 \cdot (- 4 (x \cdot x)) - 96 x \cdot 16 + 3 x^{2} (- 4 x) + 3 x^{2} \cdot 16$

Step 1.1.5.11.4.2

Multiply $x$ by $x$ .

$f' (x) = - 3072 x + 12288 - 96 \cdot (- 4 x^{2}) - 96 x \cdot 16 + 3 x^{2} (- 4 x) + 3 x^{2} \cdot 16$

Step 1.1.5.11.5

Multiply $- 96$ by $- 4$ .

$f' (x) = - 3072 x + 12288 + 384 x^{2} - 96 x \cdot 16 + 3 x^{2} (- 4 x) + 3 x^{2} \cdot 16$

Step 1.1.5.11.6

Multiply $16$ by $- 96$ .

$f' (x) = - 3072 x + 12288 + 384 x^{2} - 1536 x + 3 x^{2} (- 4 x) + 3 x^{2} \cdot 16$

Step 1.1.5.11.7

Rewrite using the commutative property of multiplication.

$f' (x) = - 3072 x + 12288 + 384 x^{2} - 1536 x + 3 \cdot (- 4 x^{2} x) + 3 x^{2} \cdot 16$

Step 1.1.5.11.8

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

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Step 1.1.5.11.8.1

Move $x$ .

$f' (x) = - 3072 x + 12288 + 384 x^{2} - 1536 x + 3 \cdot (- 4 (x \cdot x^{2})) + 3 x^{2} \cdot 16$

Step 1.1.5.11.8.2

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

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Step 1.1.5.11.8.2.1

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

$f' (x) = - 3072 x + 12288 + 384 x^{2} - 1536 x + 3 \cdot (- 4 (x \cdot x^{2})) + 3 x^{2} \cdot 16$

Step 1.1.5.11.8.2.2

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

$f' (x) = - 3072 x + 12288 + 384 x^{2} - 1536 x + 3 \cdot (- 4 x^{1 + 2}) + 3 x^{2} \cdot 16$

Step 1.1.5.11.8.3

Add $1$ and $2$ .

$f' (x) = - 3072 x + 12288 + 384 x^{2} - 1536 x + 3 \cdot (- 4 x^{3}) + 3 x^{2} \cdot 16$

Step 1.1.5.11.9

Multiply $3$ by $- 4$ .

$f' (x) = - 3072 x + 12288 + 384 x^{2} - 1536 x - 12 x^{3} + 3 x^{2} \cdot 16$

Step 1.1.5.11.10

Multiply $16$ by $3$ .

$f' (x) = - 3072 x + 12288 + 384 x^{2} - 1536 x - 12 x^{3} + 48 x^{2}$

Step 1.1.5.12

Subtract $1536 x$ from $- 3072 x$ .

$f' (x) = - 4608 x + 12288 + 384 x^{2} - 12 x^{3} + 48 x^{2}$

Step 1.1.5.13

Add $384 x^{2}$ and $48 x^{2}$ .

$f' (x) = - 4608 x + 12288 - 12 x^{3} + 432 x^{2}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- 4608 x + 12288 - 12 x^{3} + 432 x^{2}$ .

$- 4608 x + 12288 - 12 x^{3} + 432 x^{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- 4608 x + 12288 - 12 x^{3} + 432 x^{2} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- 4608 x + 12288 - 12 x^{3} + 432 x^{2} = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $12$ out of $- 4608 x + 12288 - 12 x^{3} + 432 x^{2}$ .

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Step 2.2.1.1

Factor $12$ out of $- 4608 x$ .

$12 (- 384 x) + 12288 - 12 x^{3} + 432 x^{2} = 0$

Step 2.2.1.2

Factor $12$ out of $12288$ .

$12 (- 384 x) + 12 (1024) - 12 x^{3} + 432 x^{2} = 0$

Step 2.2.1.3

Factor $12$ out of $- 12 x^{3}$ .

$12 (- 384 x) + 12 (1024) + 12 (- x^{3}) + 432 x^{2} = 0$

Step 2.2.1.4

Factor $12$ out of $432 x^{2}$ .

$12 (- 384 x) + 12 (1024) + 12 (- x^{3}) + 12 (36 x^{2}) = 0$

Step 2.2.1.5

Factor $12$ out of $12 (- 384 x) + 12 (1024)$ .

$12 (- 384 x + 1024) + 12 (- x^{3}) + 12 (36 x^{2}) = 0$

Step 2.2.1.6

Factor $12$ out of $12 (- 384 x + 1024) + 12 (- x^{3})$ .

$12 (- 384 x + 1024 - x^{3}) + 12 (36 x^{2}) = 0$

Step 2.2.1.7

Factor $12$ out of $12 (- 384 x + 1024 - x^{3}) + 12 (36 x^{2})$ .

$12 (- 384 x + 1024 - x^{3} + 36 x^{2}) = 0$

Step 2.2.2

Reorder terms.

$12 (- x^{3} + 36 x^{2} - 384 x + 1024) = 0$

Step 2.2.3

Factor.

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Step 2.2.3.1

Factor $- x^{3} + 36 x^{2} - 384 x + 1024$ using the rational roots test.

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Step 2.2.3.1.1

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.

$p = \pm 1, \pm 1024, \pm 2, \pm 512, \pm 4, \pm 256, \pm 8, \pm 128, \pm 16, \pm 64, \pm 32$

$q = \pm 1$

Step 2.2.3.1.2

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

$\pm 1, \pm 1024, \pm 2, \pm 512, \pm 4, \pm 256, \pm 8, \pm 128, \pm 16, \pm 64, \pm 32$

Step 2.2.3.1.3

Substitute $4$ and simplify the expression. In this case, the expression is equal to $0$ so $4$ is a root of the polynomial.

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Step 2.2.3.1.3.1

Substitute $4$ into the polynomial.

$- 4^{3} + 36 \cdot 4^{2} - 384 \cdot 4 + 1024$

Step 2.2.3.1.3.2

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

$- 1 \cdot 64 + 36 \cdot 4^{2} - 384 \cdot 4 + 1024$

Step 2.2.3.1.3.3

Multiply $- 1$ by $64$ .

$- 64 + 36 \cdot 4^{2} - 384 \cdot 4 + 1024$

Step 2.2.3.1.3.4

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

$- 64 + 36 \cdot 16 - 384 \cdot 4 + 1024$

Step 2.2.3.1.3.5

Multiply $36$ by $16$ .

$- 64 + 576 - 384 \cdot 4 + 1024$

Step 2.2.3.1.3.6

Add $- 64$ and $576$ .

$512 - 384 \cdot 4 + 1024$

Step 2.2.3.1.3.7

Multiply $- 384$ by $4$ .

$512 - 1536 + 1024$

Step 2.2.3.1.3.8

Subtract $1536$ from $512$ .

$- 1024 + 1024$

Step 2.2.3.1.3.9

Add $- 1024$ and $1024$ .

$0$

Step 2.2.3.1.4

Since $4$ is a known root, divide the polynomial by $x - 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- x^{3} + 36 x^{2} - 384 x + 1024}{x - 4}$

Step 2.2.3.1.5

Divide $- x^{3} + 36 x^{2} - 384 x + 1024$ by $x - 4$ .

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Step 2.2.3.1.5.1

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

$x$

$4$

$x^{3}$

$36 x^{2}$

$384 x$

$1024$

Step 2.2.3.1.5.2

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

				-	$x^{2}$
	$x$	-	$4$	-	$x^{3}$	+	$36 x^{2}$	-	$384 x$	+	$1024$

Step 2.2.3.1.5.3

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	-	$4$	-	$x^{3}$	+	$36 x^{2}$	-	$384 x$	+	$1024$
			-	$x^{3}$	+	$4 x^{2}$

Step 2.2.3.1.5.4

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

			-	$x^{2}$
$x$	-	$4$	-	$x^{3}$	+	$36 x^{2}$	-	$384 x$	+	$1024$
			+	$x^{3}$	-	$4 x^{2}$

Step 2.2.3.1.5.5

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

			-	$x^{2}$
$x$	-	$4$	-	$x^{3}$	+	$36 x^{2}$	-	$384 x$	+	$1024$
			+	$x^{3}$	-	$4 x^{2}$
					+	$32 x^{2}$

Step 2.2.3.1.5.6

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

			-	$x^{2}$
$x$	-	$4$	-	$x^{3}$	+	$36 x^{2}$	-	$384 x$	+	$1024$
			+	$x^{3}$	-	$4 x^{2}$
					+	$32 x^{2}$	-	$384 x$

Step 2.2.3.1.5.7

Divide the highest order term in the dividend $32 x^{2}$ by the highest order term in divisor $x$ .

			-	$x^{2}$	+	$32 x$
$x$	-	$4$	-	$x^{3}$	+	$36 x^{2}$	-	$384 x$	+	$1024$
			+	$x^{3}$	-	$4 x^{2}$
					+	$32 x^{2}$	-	$384 x$

Step 2.2.3.1.5.8

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$32 x$
$x$	-	$4$	-	$x^{3}$	+	$36 x^{2}$	-	$384 x$	+	$1024$
			+	$x^{3}$	-	$4 x^{2}$
					+	$32 x^{2}$	-	$384 x$
					+	$32 x^{2}$	-	$128 x$

Step 2.2.3.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $32 x^{2} - 128 x$

			-	$x^{2}$	+	$32 x$
$x$	-	$4$	-	$x^{3}$	+	$36 x^{2}$	-	$384 x$	+	$1024$
			+	$x^{3}$	-	$4 x^{2}$
					+	$32 x^{2}$	-	$384 x$
					-	$32 x^{2}$	+	$128 x$

Step 2.2.3.1.5.10

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

			-	$x^{2}$	+	$32 x$
$x$	-	$4$	-	$x^{3}$	+	$36 x^{2}$	-	$384 x$	+	$1024$
			+	$x^{3}$	-	$4 x^{2}$
					+	$32 x^{2}$	-	$384 x$
					-	$32 x^{2}$	+	$128 x$
							-	$256 x$

Step 2.2.3.1.5.11

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

			-	$x^{2}$	+	$32 x$
$x$	-	$4$	-	$x^{3}$	+	$36 x^{2}$	-	$384 x$	+	$1024$
			+	$x^{3}$	-	$4 x^{2}$
					+	$32 x^{2}$	-	$384 x$
					-	$32 x^{2}$	+	$128 x$
							-	$256 x$	+	$1024$

Step 2.2.3.1.5.12

Divide the highest order term in the dividend $- 256 x$ by the highest order term in divisor $x$ .

			-	$x^{2}$	+	$32 x$	-	$256$
$x$	-	$4$	-	$x^{3}$	+	$36 x^{2}$	-	$384 x$	+	$1024$
			+	$x^{3}$	-	$4 x^{2}$
					+	$32 x^{2}$	-	$384 x$
					-	$32 x^{2}$	+	$128 x$
							-	$256 x$	+	$1024$

Step 2.2.3.1.5.13

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$32 x$	-	$256$
$x$	-	$4$	-	$x^{3}$	+	$36 x^{2}$	-	$384 x$	+	$1024$
			+	$x^{3}$	-	$4 x^{2}$
					+	$32 x^{2}$	-	$384 x$
					-	$32 x^{2}$	+	$128 x$
							-	$256 x$	+	$1024$
							-	$256 x$	+	$1024$

Step 2.2.3.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 256 x + 1024$

			-	$x^{2}$	+	$32 x$	-	$256$
$x$	-	$4$	-	$x^{3}$	+	$36 x^{2}$	-	$384 x$	+	$1024$
			+	$x^{3}$	-	$4 x^{2}$
					+	$32 x^{2}$	-	$384 x$
					-	$32 x^{2}$	+	$128 x$
							-	$256 x$	+	$1024$
							+	$256 x$	-	$1024$

Step 2.2.3.1.5.15

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

			-	$x^{2}$	+	$32 x$	-	$256$
$x$	-	$4$	-	$x^{3}$	+	$36 x^{2}$	-	$384 x$	+	$1024$
			+	$x^{3}$	-	$4 x^{2}$
					+	$32 x^{2}$	-	$384 x$
					-	$32 x^{2}$	+	$128 x$
							-	$256 x$	+	$1024$
							+	$256 x$	-	$1024$
										$0$

Step 2.2.3.1.5.16

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

$- x^{2} + 32 x - 256$

Step 2.2.3.1.6

Write $- x^{3} + 36 x^{2} - 384 x + 1024$ as a set of factors.

$12 ((x - 4) (- x^{2} + 32 x - 256)) = 0$

Step 2.2.3.2

Remove unnecessary parentheses.

$12 (x - 4) (- x^{2} + 32 x - 256) = 0$

Step 2.2.4

Factor.

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Step 2.2.4.1

Factor by grouping.

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Step 2.2.4.1.1

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 $a \cdot c = - 1 \cdot - 256 = 256$ and whose sum is $b = 32$ .

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Step 2.2.4.1.1.1

Factor $32$ out of $32 x$ .

$12 (x - 4) (- x^{2} + 32 (x) - 256) = 0$

Step 2.2.4.1.1.2

Rewrite $32$ as $16$ plus $16$

$12 (x - 4) (- x^{2} + (16 + 16) x - 256) = 0$

Step 2.2.4.1.1.3

Apply the distributive property.

$12 (x - 4) (- x^{2} + 16 x + 16 x - 256) = 0$

Step 2.2.4.1.2

Factor out the greatest common factor from each group.

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Step 2.2.4.1.2.1

Group the first two terms and the last two terms.

$12 (x - 4) ((- x^{2} + 16 x) + 16 x - 256) = 0$

Step 2.2.4.1.2.2

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

$12 (x - 4) (x (- x + 16) - 16 (- x + 16)) = 0$

Step 2.2.4.1.3

Factor the polynomial by factoring out the greatest common factor, $- x + 16$ .

$12 (x - 4) ((- x + 16) (x - 16)) = 0$

Step 2.2.4.2

Remove unnecessary parentheses.

$12 (x - 4) (- x + 16) (x - 16) = 0$

Step 2.2.5

Combine exponents.

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Step 2.2.5.1

Factor $- 1$ out of $- x$ .

$12 (x - 4) (- (x) + 16) (x - 16) = 0$

Step 2.2.5.2

Rewrite $16$ as $- 1 (- 16)$ .

$12 (x - 4) (- (x) - 1 \cdot - 16) (x - 16) = 0$

Step 2.2.5.3

Factor $- 1$ out of $- (x) - 1 (- 16)$ .

$12 (x - 4) (- (x - 16)) (x - 16) = 0$

Step 2.2.5.4

Rewrite $- (x - 16)$ as $- 1 (x - 16)$ .

$12 (x - 4) (- 1 (x - 16)) (x - 16) = 0$

Step 2.2.5.5

Remove parentheses.

$12 (x - 4) \cdot (- 1 (x - 16) (x - 16)) = 0$

Step 2.2.5.6

Raise $x - 16$ to the power of $1$ .

$12 (x - 4) \cdot (- 1 ((x - 16) (x - 16))) = 0$

Step 2.2.5.7

Raise $x - 16$ to the power of $1$ .

$12 (x - 4) \cdot (- 1 ((x - 16) (x - 16))) = 0$

Step 2.2.5.8

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

$12 (x - 4) \cdot (- 1 {(x - 16)}^{1 + 1}) = 0$

Step 2.2.5.9

Add $1$ and $1$ .

$12 (x - 4) \cdot (- 1 {(x - 16)}^{2}) = 0$

Step 2.2.5.10

Multiply $- 1$ by $12$ .

$- 12 (x - 4) {(x - 16)}^{2} = 0$

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 $0$ .

$x - 4 = 0$

${(x - 16)}^{2} = 0$

Step 2.4

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

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Step 2.4.1

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

$x - 4 = 0$

Step 2.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.5

Set ${(x - 16)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set ${(x - 16)}^{2}$ equal to $0$ .

${(x - 16)}^{2} = 0$

Step 2.5.2

Solve ${(x - 16)}^{2} = 0$ for $x$ .

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Step 2.5.2.1

Set the $x - 16$ equal to $0$ .

$x - 16 = 0$

Step 2.5.2.2

Add $16$ to both sides of the equation.

$x = 16$

Step 2.6

The final solution is all the values that make $- 12 (x - 4) {(x - 16)}^{2} = 0$ true.

$x = 4, 16$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $3 x {(16 - x)}^{3}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 4$ .

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Step 4.1.1

Substitute $4$ for $x$ .

$3 (4) {(16 - (4))}^{3}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $3$ by $4$ .

$12 {(16 - (4))}^{3}$

Step 4.1.2.2

Multiply $- 1$ by $4$ .

$12 {(16 - 4)}^{3}$

Step 4.1.2.3

Subtract $4$ from $16$ .

$12 \cdot 12^{3}$

Step 4.1.2.4

Multiply $12$ by $12^{3}$ by adding the exponents.

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Step 4.1.2.4.1

Multiply $12$ by $12^{3}$ .

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Step 4.1.2.4.1.1

Raise $12$ to the power of $1$ .

$12^{1} \cdot 12^{3}$

Step 4.1.2.4.1.2

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

$12^{1 + 3}$

Step 4.1.2.4.2

Add $1$ and $3$ .

$12^{4}$

Step 4.1.2.5

Raise $12$ to the power of $4$ .

$20736$

Step 4.2

Evaluate at $x = 16$ .

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Step 4.2.1

Substitute $16$ for $x$ .

$3 (16) {(16 - (16))}^{3}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Multiply $3$ by $16$ .

$48 {(16 - (16))}^{3}$

Step 4.2.2.2

Multiply $- 1$ by $16$ .

$48 {(16 - 16)}^{3}$

Step 4.2.2.3

Subtract $16$ from $16$ .

$48 \cdot 0^{3}$

Step 4.2.2.4

Raising $0$ to any positive power yields $0$ .

$48 \cdot 0$

Step 4.2.2.5

Multiply $48$ by $0$ .

$0$

Step 4.3

List all of the points.

$(4, 20736), (16, 0)$

Step 5