Calculus Examples

$f (x) = x^{2} {(4 - 5 x)}^{3}$

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

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 $g (x) = {(4 - 5 x)}^{3}$ .

$x^{2} \frac{d}{d x} [{(4 - 5 x)}^{3}] + {(4 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

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 $g (x) = 4 - 5 x$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $4 - 5 x$ .

$x^{2} (\frac{d}{d u} [u^{3}] \frac{d}{d x} [4 - 5 x]) + {(4 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

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 $n = 3$ .

$x^{2} (3 u^{2} \frac{d}{d x} [4 - 5 x]) + {(4 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.2.3

Replace all occurrences of $u$ with $4 - 5 x$ .

$x^{2} (3 {(4 - 5 x)}^{2} \frac{d}{d x} [4 - 5 x]) + {(4 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

By the Sum Rule, the derivative of $4 - 5 x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 5 x]$ .

$x^{2} (3 {(4 - 5 x)}^{2} (\frac{d}{d x} [4] + \frac{d}{d x} [- 5 x])) + {(4 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.3.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$x^{2} (3 {(4 - 5 x)}^{2} (0 + \frac{d}{d x} [- 5 x])) + {(4 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.3.3

Add $0$ and $\frac{d}{d x} [- 5 x]$ .

$x^{2} (3 {(4 - 5 x)}^{2} \frac{d}{d x} [- 5 x]) + {(4 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.3.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$x^{2} (3 {(4 - 5 x)}^{2} (- 5 \frac{d}{d x} [x])) + {(4 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.3.5

Multiply $- 5$ by $3$ .

$x^{2} (- 15 {(4 - 5 x)}^{2} \frac{d}{d x} [x]) + {(4 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.3.6

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

$x^{2} (- 15 {(4 - 5 x)}^{2} \cdot 1) + {(4 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.3.7

Multiply $- 15$ by $1$ .

$x^{2} (- 15 {(4 - 5 x)}^{2}) + {(4 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.1.3.8

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

$x^{2} (- 15 {(4 - 5 x)}^{2}) + {(4 - 5 x)}^{3} (2 x)$

Step 1.1.3.9

Move $2$ to the left of ${(4 - 5 x)}^{3}$ .

$x^{2} (- 15 {(4 - 5 x)}^{2}) + 2 \cdot {(4 - 5 x)}^{3} x$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Factor $x {(4 - 5 x)}^{2}$ out of $x^{2} (- 15 {(4 - 5 x)}^{2}) + 2 {(4 - 5 x)}^{3} x$ .

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Step 1.1.4.1.1

Factor $x {(4 - 5 x)}^{2}$ out of $x^{2} (- 15 {(4 - 5 x)}^{2})$ .

$x {(4 - 5 x)}^{2} (x \cdot (- 15)) + 2 {(4 - 5 x)}^{3} x$

Step 1.1.4.1.2

Factor $x {(4 - 5 x)}^{2}$ out of $2 {(4 - 5 x)}^{3} x$ .

$x {(4 - 5 x)}^{2} (x \cdot (- 15)) + x {(4 - 5 x)}^{2} (2 (4 - 5 x))$

Step 1.1.4.1.3

Factor $x {(4 - 5 x)}^{2}$ out of $x {(4 - 5 x)}^{2} (x \cdot (- 15)) + x {(4 - 5 x)}^{2} (2 (4 - 5 x))$ .

$x {(4 - 5 x)}^{2} (x \cdot (- 15) + 2 (4 - 5 x))$

Step 1.1.4.2

Move $- 15$ to the left of $x$ .

$x {(4 - 5 x)}^{2} (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.3

Rewrite ${(4 - 5 x)}^{2}$ as $(4 - 5 x) (4 - 5 x)$ .

$x ((4 - 5 x) (4 - 5 x)) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.4

Expand $(4 - 5 x) (4 - 5 x)$ using the FOIL Method.

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Step 1.1.4.4.1

Apply the distributive property.

$x (4 (4 - 5 x) - 5 x (4 - 5 x)) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.4.2

Apply the distributive property.

$x (4 \cdot 4 + 4 (- 5 x) - 5 x (4 - 5 x)) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.4.3

Apply the distributive property.

$x (4 \cdot 4 + 4 (- 5 x) - 5 x \cdot 4 - 5 x (- 5 x)) (- 15 x + 2 (4 - 5 x))$

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 $4$ by $4$ .

$x (16 + 4 (- 5 x) - 5 x \cdot 4 - 5 x (- 5 x)) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.5.1.2

Multiply $- 5$ by $4$ .

$x (16 - 20 x - 5 x \cdot 4 - 5 x (- 5 x)) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.5.1.3

Multiply $4$ by $- 5$ .

$x (16 - 20 x - 20 x - 5 x (- 5 x)) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.5.1.4

Rewrite using the commutative property of multiplication.

$x (16 - 20 x - 20 x - 5 \cdot - 5 x \cdot x) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.5.1.5

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

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Step 1.1.4.5.1.5.1

Move $x$ .

$x (16 - 20 x - 20 x - 5 \cdot - 5 (x \cdot x)) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.5.1.5.2

Multiply $x$ by $x$ .

$x (16 - 20 x - 20 x - 5 \cdot - 5 x^{2}) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.5.1.6

Multiply $- 5$ by $- 5$ .

$x (16 - 20 x - 20 x + 25 x^{2}) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.5.2

Subtract $20 x$ from $- 20 x$ .

$x (16 - 40 x + 25 x^{2}) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.6

Apply the distributive property.

$(x \cdot 16 + x (- 40 x) + x (25 x^{2})) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.7

Simplify.

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Step 1.1.4.7.1

Move $16$ to the left of $x$ .

$(16 \cdot x + x (- 40 x) + x (25 x^{2})) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.7.2

Rewrite using the commutative property of multiplication.

$(16 \cdot x - 40 x \cdot x + x (25 x^{2})) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.7.3

Rewrite using the commutative property of multiplication.

$(16 \cdot x - 40 x \cdot x + 25 x \cdot x^{2}) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.8

Simplify each term.

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Step 1.1.4.8.1

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

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Step 1.1.4.8.1.1

Move $x$ .

$(16 x - 40 (x \cdot x) + 25 x \cdot x^{2}) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.8.1.2

Multiply $x$ by $x$ .

$(16 x - 40 x^{2} + 25 x \cdot x^{2}) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.8.2

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

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Step 1.1.4.8.2.1

Move $x^{2}$ .

$(16 x - 40 x^{2} + 25 (x^{2} x)) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.8.2.2

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

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Step 1.1.4.8.2.2.1

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

$(16 x - 40 x^{2} + 25 (x^{2} x^{1})) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.8.2.2.2

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

$(16 x - 40 x^{2} + 25 x^{2 + 1}) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.8.2.3

Add $2$ and $1$ .

$(16 x - 40 x^{2} + 25 x^{3}) (- 15 x + 2 (4 - 5 x))$

Step 1.1.4.9

Simplify each term.

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Step 1.1.4.9.1

Apply the distributive property.

$(16 x - 40 x^{2} + 25 x^{3}) (- 15 x + 2 \cdot 4 + 2 (- 5 x))$

Step 1.1.4.9.2

Multiply $2$ by $4$ .

$(16 x - 40 x^{2} + 25 x^{3}) (- 15 x + 8 + 2 (- 5 x))$

Step 1.1.4.9.3

Multiply $- 5$ by $2$ .

$(16 x - 40 x^{2} + 25 x^{3}) (- 15 x + 8 - 10 x)$

Step 1.1.4.10

Subtract $10 x$ from $- 15 x$ .

$(16 x - 40 x^{2} + 25 x^{3}) (- 25 x + 8)$

Step 1.1.4.11

Expand $(16 x - 40 x^{2} + 25 x^{3}) (- 25 x + 8)$ by multiplying each term in the first expression by each term in the second expression.

$16 x (- 25 x) + 16 x \cdot 8 - 40 x^{2} (- 25 x) - 40 x^{2} \cdot 8 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 8$

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.

$16 \cdot - 25 x \cdot x + 16 x \cdot 8 - 40 x^{2} (- 25 x) - 40 x^{2} \cdot 8 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 8$

Step 1.1.4.12.2

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

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Step 1.1.4.12.2.1

Move $x$ .

$16 \cdot - 25 (x \cdot x) + 16 x \cdot 8 - 40 x^{2} (- 25 x) - 40 x^{2} \cdot 8 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 8$

Step 1.1.4.12.2.2

Multiply $x$ by $x$ .

$16 \cdot - 25 x^{2} + 16 x \cdot 8 - 40 x^{2} (- 25 x) - 40 x^{2} \cdot 8 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 8$

Step 1.1.4.12.3

Multiply $16$ by $- 25$ .

$- 400 x^{2} + 16 x \cdot 8 - 40 x^{2} (- 25 x) - 40 x^{2} \cdot 8 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 8$

Step 1.1.4.12.4

Multiply $8$ by $16$ .

$- 400 x^{2} + 128 x - 40 x^{2} (- 25 x) - 40 x^{2} \cdot 8 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 8$

Step 1.1.4.12.5

Rewrite using the commutative property of multiplication.

$- 400 x^{2} + 128 x - 40 \cdot - 25 x^{2} x - 40 x^{2} \cdot 8 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 8$

Step 1.1.4.12.6

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

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Step 1.1.4.12.6.1

Move $x$ .

$- 400 x^{2} + 128 x - 40 \cdot - 25 (x \cdot x^{2}) - 40 x^{2} \cdot 8 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 8$

Step 1.1.4.12.6.2

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

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Step 1.1.4.12.6.2.1

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

$- 400 x^{2} + 128 x - 40 \cdot - 25 (x^{1} x^{2}) - 40 x^{2} \cdot 8 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 8$

Step 1.1.4.12.6.2.2

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

$- 400 x^{2} + 128 x - 40 \cdot - 25 x^{1 + 2} - 40 x^{2} \cdot 8 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 8$

Step 1.1.4.12.6.3

Add $1$ and $2$ .

$- 400 x^{2} + 128 x - 40 \cdot - 25 x^{3} - 40 x^{2} \cdot 8 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 8$

Step 1.1.4.12.7

Multiply $- 40$ by $- 25$ .

$- 400 x^{2} + 128 x + 1000 x^{3} - 40 x^{2} \cdot 8 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 8$

Step 1.1.4.12.8

Multiply $8$ by $- 40$ .

$- 400 x^{2} + 128 x + 1000 x^{3} - 320 x^{2} + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 8$

Step 1.1.4.12.9

Rewrite using the commutative property of multiplication.

$- 400 x^{2} + 128 x + 1000 x^{3} - 320 x^{2} + 25 \cdot - 25 x^{3} x + 25 x^{3} \cdot 8$

Step 1.1.4.12.10

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

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Step 1.1.4.12.10.1

Move $x$ .

$- 400 x^{2} + 128 x + 1000 x^{3} - 320 x^{2} + 25 \cdot - 25 (x \cdot x^{3}) + 25 x^{3} \cdot 8$

Step 1.1.4.12.10.2

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

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Step 1.1.4.12.10.2.1

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

$- 400 x^{2} + 128 x + 1000 x^{3} - 320 x^{2} + 25 \cdot - 25 (x^{1} x^{3}) + 25 x^{3} \cdot 8$

Step 1.1.4.12.10.2.2

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

$- 400 x^{2} + 128 x + 1000 x^{3} - 320 x^{2} + 25 \cdot - 25 x^{1 + 3} + 25 x^{3} \cdot 8$

Step 1.1.4.12.10.3

Add $1$ and $3$ .

$- 400 x^{2} + 128 x + 1000 x^{3} - 320 x^{2} + 25 \cdot - 25 x^{4} + 25 x^{3} \cdot 8$

Step 1.1.4.12.11

Multiply $25$ by $- 25$ .

$- 400 x^{2} + 128 x + 1000 x^{3} - 320 x^{2} - 625 x^{4} + 25 x^{3} \cdot 8$

Step 1.1.4.12.12

Multiply $8$ by $25$ .

$- 400 x^{2} + 128 x + 1000 x^{3} - 320 x^{2} - 625 x^{4} + 200 x^{3}$

Step 1.1.4.13

Subtract $320 x^{2}$ from $- 400 x^{2}$ .

$- 720 x^{2} + 128 x + 1000 x^{3} - 625 x^{4} + 200 x^{3}$

Step 1.1.4.14

Add $1000 x^{3}$ and $200 x^{3}$ .

$f' (x) = - 720 x^{2} + 128 x - 625 x^{4} + 1200 x^{3}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- 720 x^{2} + 128 x - 625 x^{4} + 1200 x^{3}$ .

$- 720 x^{2} + 128 x - 625 x^{4} + 1200 x^{3}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- 720 x^{2} + 128 x - 625 x^{4} + 1200 x^{3} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- 720 x^{2} + 128 x - 625 x^{4} + 1200 x^{3} = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $x$ out of $- 720 x^{2} + 128 x - 625 x^{4} + 1200 x^{3}$ .

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Step 2.2.1.1

Factor $x$ out of $- 720 x^{2}$ .

$x (- 720 x) + 128 x - 625 x^{4} + 1200 x^{3} = 0$

Step 2.2.1.2

Factor $x$ out of $128 x$ .

$x (- 720 x) + x \cdot 128 - 625 x^{4} + 1200 x^{3} = 0$

Step 2.2.1.3

Factor $x$ out of $- 625 x^{4}$ .

$x (- 720 x) + x \cdot 128 + x (- 625 x^{3}) + 1200 x^{3} = 0$

Step 2.2.1.4

Factor $x$ out of $1200 x^{3}$ .

$x (- 720 x) + x \cdot 128 + x (- 625 x^{3}) + x (1200 x^{2}) = 0$

Step 2.2.1.5

Factor $x$ out of $x (- 720 x) + x \cdot 128$ .

$x (- 720 x + 128) + x (- 625 x^{3}) + x (1200 x^{2}) = 0$

Step 2.2.1.6

Factor $x$ out of $x (- 720 x + 128) + x (- 625 x^{3})$ .

$x (- 720 x + 128 - 625 x^{3}) + x (1200 x^{2}) = 0$

Step 2.2.1.7

Factor $x$ out of $x (- 720 x + 128 - 625 x^{3}) + x (1200 x^{2})$ .

$x (- 720 x + 128 - 625 x^{3} + 1200 x^{2}) = 0$

Step 2.2.2

Reorder terms.

$x (- 625 x^{3} + 1200 x^{2} - 720 x + 128) = 0$

Step 2.2.3

Factor.

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Step 2.2.3.1

Factor $- 625 x^{3} + 1200 x^{2} - 720 x + 128$ 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.

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

$q = \pm 1, \pm 625, \pm 5, \pm 125, \pm 25$

Step 2.2.3.1.2

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

$\pm 1, \pm 0.0016, \pm 0.2, \pm 0.008, \pm 0.04, \pm 128, \pm 0.2048, \pm 25.6, \pm 1.024, \pm 5.12, \pm 2, \pm 0.0032, \pm 0.4, \pm 0.016, \pm 0.08, \pm 64, \pm 0.1024, \pm 12.8, \pm 0.512, \pm 2.56, \pm 4, \pm 0.0064, \pm 0.8, \pm 0.032, \pm 0.16, \pm 32, \pm 0.0512, \pm 6.4, \pm 0.256, \pm 1.28, \pm 8, \pm 0.0128, \pm 1.6, \pm 0.064, \pm 0.32, \pm 16, \pm 0.0256, \pm 3.2, \pm 0.128, \pm 0.64$

Step 2.2.3.1.3

Substitute $0.8$ and simplify the expression. In this case, the expression is equal to $0$ so $0.8$ is a root of the polynomial.

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Step 2.2.3.1.3.1

Substitute $0.8$ into the polynomial.

$- 625 \cdot {0.8}^{3} + 1200 \cdot {0.8}^{2} - 720 \cdot 0.8 + 128$

Step 2.2.3.1.3.2

Raise $0.8$ to the power of $3$ .

$- 625 \cdot 0.512 + 1200 \cdot {0.8}^{2} - 720 \cdot 0.8 + 128$

Step 2.2.3.1.3.3

Multiply $- 625$ by $0.512$ .

$- 320 + 1200 \cdot {0.8}^{2} - 720 \cdot 0.8 + 128$

Step 2.2.3.1.3.4

Raise $0.8$ to the power of $2$ .

$- 320 + 1200 \cdot 0.64 - 720 \cdot 0.8 + 128$

Step 2.2.3.1.3.5

Multiply $1200$ by $0.64$ .

$- 320 + 768 - 720 \cdot 0.8 + 128$

Step 2.2.3.1.3.6

Add $- 320$ and $768$ .

$448 - 720 \cdot 0.8 + 128$

Step 2.2.3.1.3.7

Multiply $- 720$ by $0.8$ .

$448 - 576 + 128$

Step 2.2.3.1.3.8

Subtract $576$ from $448$ .

$- 128 + 128$

Step 2.2.3.1.3.9

Add $- 128$ and $128$ .

$0$

Step 2.2.3.1.4

Since $0.8$ is a known root, divide the polynomial by $5 x - 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- 625 x^{3} + 1200 x^{2} - 720 x + 128}{5 x - 4}$

Step 2.2.3.1.5

Divide $- 625 x^{3} + 1200 x^{2} - 720 x + 128$ by $5 x - 4$ .

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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 $0$ .

$5 x$

$4$

$625 x^{3}$

$1200 x^{2}$

$720 x$

$128$

Step 2.2.3.1.5.2

Divide the highest order term in the dividend $- 625 x^{3}$ by the highest order term in divisor $5 x$ .

				-	$125 x^{2}$
	$5 x$	-	$4$	-	$625 x^{3}$	+	$1200 x^{2}$	-	$720 x$	+	$128$

Step 2.2.3.1.5.3

Multiply the new quotient term by the divisor.

			-	$125 x^{2}$
$5 x$	-	$4$	-	$625 x^{3}$	+	$1200 x^{2}$	-	$720 x$	+	$128$
			-	$625 x^{3}$	+	$500 x^{2}$

Step 2.2.3.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $- 625 x^{3} + 500 x^{2}$

			-	$125 x^{2}$
$5 x$	-	$4$	-	$625 x^{3}$	+	$1200 x^{2}$	-	$720 x$	+	$128$
			+	$625 x^{3}$	-	$500 x^{2}$

Step 2.2.3.1.5.5

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

			-	$125 x^{2}$
$5 x$	-	$4$	-	$625 x^{3}$	+	$1200 x^{2}$	-	$720 x$	+	$128$
			+	$625 x^{3}$	-	$500 x^{2}$
					+	$700 x^{2}$

Step 2.2.3.1.5.6

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

			-	$125 x^{2}$
$5 x$	-	$4$	-	$625 x^{3}$	+	$1200 x^{2}$	-	$720 x$	+	$128$
			+	$625 x^{3}$	-	$500 x^{2}$
					+	$700 x^{2}$	-	$720 x$

Step 2.2.3.1.5.7

Divide the highest order term in the dividend $700 x^{2}$ by the highest order term in divisor $5 x$ .

			-	$125 x^{2}$	+	$140 x$
$5 x$	-	$4$	-	$625 x^{3}$	+	$1200 x^{2}$	-	$720 x$	+	$128$
			+	$625 x^{3}$	-	$500 x^{2}$
					+	$700 x^{2}$	-	$720 x$

Step 2.2.3.1.5.8

Multiply the new quotient term by the divisor.

			-	$125 x^{2}$	+	$140 x$
$5 x$	-	$4$	-	$625 x^{3}$	+	$1200 x^{2}$	-	$720 x$	+	$128$
			+	$625 x^{3}$	-	$500 x^{2}$
					+	$700 x^{2}$	-	$720 x$
					+	$700 x^{2}$	-	$560 x$

Step 2.2.3.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $700 x^{2} - 560 x$

			-	$125 x^{2}$	+	$140 x$
$5 x$	-	$4$	-	$625 x^{3}$	+	$1200 x^{2}$	-	$720 x$	+	$128$
			+	$625 x^{3}$	-	$500 x^{2}$
					+	$700 x^{2}$	-	$720 x$
					-	$700 x^{2}$	+	$560 x$

Step 2.2.3.1.5.10

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

			-	$125 x^{2}$	+	$140 x$
$5 x$	-	$4$	-	$625 x^{3}$	+	$1200 x^{2}$	-	$720 x$	+	$128$
			+	$625 x^{3}$	-	$500 x^{2}$
					+	$700 x^{2}$	-	$720 x$
					-	$700 x^{2}$	+	$560 x$
							-	$160 x$

Step 2.2.3.1.5.11

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

			-	$125 x^{2}$	+	$140 x$
$5 x$	-	$4$	-	$625 x^{3}$	+	$1200 x^{2}$	-	$720 x$	+	$128$
			+	$625 x^{3}$	-	$500 x^{2}$
					+	$700 x^{2}$	-	$720 x$
					-	$700 x^{2}$	+	$560 x$
							-	$160 x$	+	$128$

Step 2.2.3.1.5.12

Divide the highest order term in the dividend $- 160 x$ by the highest order term in divisor $5 x$ .

			-	$125 x^{2}$	+	$140 x$	-	$32$
$5 x$	-	$4$	-	$625 x^{3}$	+	$1200 x^{2}$	-	$720 x$	+	$128$
			+	$625 x^{3}$	-	$500 x^{2}$
					+	$700 x^{2}$	-	$720 x$
					-	$700 x^{2}$	+	$560 x$
							-	$160 x$	+	$128$

Step 2.2.3.1.5.13

Multiply the new quotient term by the divisor.

			-	$125 x^{2}$	+	$140 x$	-	$32$
$5 x$	-	$4$	-	$625 x^{3}$	+	$1200 x^{2}$	-	$720 x$	+	$128$
			+	$625 x^{3}$	-	$500 x^{2}$
					+	$700 x^{2}$	-	$720 x$
					-	$700 x^{2}$	+	$560 x$
							-	$160 x$	+	$128$
							-	$160 x$	+	$128$

Step 2.2.3.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 160 x + 128$

			-	$125 x^{2}$	+	$140 x$	-	$32$
$5 x$	-	$4$	-	$625 x^{3}$	+	$1200 x^{2}$	-	$720 x$	+	$128$
			+	$625 x^{3}$	-	$500 x^{2}$
					+	$700 x^{2}$	-	$720 x$
					-	$700 x^{2}$	+	$560 x$
							-	$160 x$	+	$128$
							+	$160 x$	-	$128$

Step 2.2.3.1.5.15

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

			-	$125 x^{2}$	+	$140 x$	-	$32$
$5 x$	-	$4$	-	$625 x^{3}$	+	$1200 x^{2}$	-	$720 x$	+	$128$
			+	$625 x^{3}$	-	$500 x^{2}$
					+	$700 x^{2}$	-	$720 x$
					-	$700 x^{2}$	+	$560 x$
							-	$160 x$	+	$128$
							+	$160 x$	-	$128$
										$0$

Step 2.2.3.1.5.16

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

$- 125 x^{2} + 140 x - 32$

Step 2.2.3.1.6

Write $- 625 x^{3} + 1200 x^{2} - 720 x + 128$ as a set of factors.

$x ((5 x - 4) (- 125 x^{2} + 140 x - 32)) = 0$

Step 2.2.3.2

Remove unnecessary parentheses.

$x (5 x - 4) (- 125 x^{2} + 140 x - 32) = 0$

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 $a \cdot c = - 125 \cdot - 32 = 4000$ and whose sum is $b = 140$ .

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Step 2.2.4.1.1.1

Factor $140$ out of $140 x$ .

$x (5 x - 4) (- 125 x^{2} + 140 (x) - 32) = 0$

Step 2.2.4.1.1.2

Rewrite $140$ as $40$ plus $100$

$x (5 x - 4) (- 125 x^{2} + (40 + 100) x - 32) = 0$

Step 2.2.4.1.1.3

Apply the distributive property.

$x (5 x - 4) (- 125 x^{2} + 40 x + 100 x - 32) = 0$

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.

$x (5 x - 4) ((- 125 x^{2} + 40 x) + 100 x - 32) = 0$

Step 2.2.4.1.2.2

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

$x (5 x - 4) (5 x (- 25 x + 8) - 4 (- 25 x + 8)) = 0$

Step 2.2.4.1.3

Factor the polynomial by factoring out the greatest common factor, $- 25 x + 8$ .

$x (5 x - 4) ((- 25 x + 8) (5 x - 4)) = 0$

Step 2.2.4.2

Remove unnecessary parentheses.

$x (5 x - 4) (- 25 x + 8) (5 x - 4) = 0$

Step 2.2.5

Combine exponents.

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Step 2.2.5.1

Raise $5 x - 4$ to the power of $1$ .

$x ((5 x - 4) (5 x - 4)) (- 25 x + 8) = 0$

Step 2.2.5.2

Raise $5 x - 4$ to the power of $1$ .

$x ((5 x - 4) (5 x - 4)) (- 25 x + 8) = 0$

Step 2.2.5.3

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

$x {(5 x - 4)}^{1 + 1} (- 25 x + 8) = 0$

Step 2.2.5.4

Add $1$ and $1$ .

$x {(5 x - 4)}^{2} (- 25 x + 8) = 0$

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 $0$ .

$x = 0$

${(5 x - 4)}^{2} = 0$

$- 25 x + 8 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set ${(5 x - 4)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set ${(5 x - 4)}^{2}$ equal to $0$ .

${(5 x - 4)}^{2} = 0$

Step 2.5.2

Solve ${(5 x - 4)}^{2} = 0$ for $x$ .

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Step 2.5.2.1

Set the $5 x - 4$ equal to $0$ .

$5 x - 4 = 0$

Step 2.5.2.2

Solve for $x$ .

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Step 2.5.2.2.1

Add $4$ to both sides of the equation.

$5 x = 4$

Step 2.5.2.2.2

Divide each term in $5 x = 4$ by $5$ and simplify.

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Step 2.5.2.2.2.1

Divide each term in $5 x = 4$ by $5$ .

$\frac{5 x}{5} = \frac{4}{5}$

Step 2.5.2.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.5.2.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{4}{5}$

Step 2.5.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{5}$

Step 2.6

Set $- 25 x + 8$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $- 25 x + 8$ equal to $0$ .

$- 25 x + 8 = 0$

Step 2.6.2

Solve $- 25 x + 8 = 0$ for $x$ .

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Step 2.6.2.1

Subtract $8$ from both sides of the equation.

$- 25 x = - 8$

Step 2.6.2.2

Divide each term in $- 25 x = - 8$ by $- 25$ and simplify.

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Step 2.6.2.2.1

Divide each term in $- 25 x = - 8$ by $- 25$ .

$\frac{- 25 x}{- 25} = \frac{- 8}{- 25}$

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 $- 25$ .

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Step 2.6.2.2.2.1.1

Cancel the common factor.

$\frac{- 25 x}{- 25} = \frac{- 8}{- 25}$

Step 2.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{- 25}$

Step 2.6.2.2.3

Simplify the right side.

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Step 2.6.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{8}{25}$

Step 2.7

The final solution is all the values that make $x {(5 x - 4)}^{2} (- 25 x + 8) = 0$ true.

$x = 0, \frac{4}{5}, \frac{8}{25}$

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 $x^{2} {(4 - 5 x)}^{3}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

${(0)}^{2} {(4 - 5 \cdot 0)}^{3}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

$0 {(4 - 5 \cdot 0)}^{3}$

Step 4.1.2.2

Multiply $- 5$ by $0$ .

$0 {(4 + 0)}^{3}$

Step 4.1.2.3

Add $4$ and $0$ .

$0 \cdot 4^{3}$

Step 4.1.2.4

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

$0 \cdot 64$

Step 4.1.2.5

Multiply $0$ by $64$ .

$0$

Step 4.2

Evaluate at $x = \frac{4}{5}$ .

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Step 4.2.1

Substitute $\frac{4}{5}$ for $x$ .

${(\frac{4}{5})}^{2} {(4 - 5 (\frac{4}{5}))}^{3}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the expression.

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Step 4.2.2.1.1

Apply the product rule to $\frac{4}{5}$ .

$\frac{4^{2}}{5^{2}} {(4 - 5 (\frac{4}{5}))}^{3}$

Step 4.2.2.1.2

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

$\frac{16}{5^{2}} {(4 - 5 (\frac{4}{5}))}^{3}$

Step 4.2.2.1.3

Raise $5$ to the power of $2$ .

$\frac{16}{25} {(4 - 5 (\frac{4}{5}))}^{3}$

Step 4.2.2.2

Simplify each term.

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Step 4.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.2.1.1

Factor $5$ out of $- 5$ .

$\frac{16}{25} {(4 + 5 (- 1) \frac{4}{5})}^{3}$

Step 4.2.2.2.1.2

Cancel the common factor.

$\frac{16}{25} {(4 + 5 \cdot - 1 \frac{4}{5})}^{3}$

Step 4.2.2.2.1.3

Rewrite the expression.

$\frac{16}{25} {(4 - 1 \cdot 4)}^{3}$

Step 4.2.2.2.2

Multiply $- 1$ by $4$ .

$\frac{16}{25} {(4 - 4)}^{3}$

Step 4.2.2.3

Simplify the expression.

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Step 4.2.2.3.1

Subtract $4$ from $4$ .

$\frac{16}{25} \cdot 0^{3}$

Step 4.2.2.3.2

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

$\frac{16}{25} \cdot 0$

Step 4.2.2.3.3

Multiply $\frac{16}{25}$ by $0$ .

$0$

Step 4.3

Evaluate at $x = \frac{8}{25}$ .

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Step 4.3.1

Substitute $\frac{8}{25}$ for $x$ .

${(\frac{8}{25})}^{2} {(4 - 5 (\frac{8}{25}))}^{3}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify the expression.

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Step 4.3.2.1.1

Apply the product rule to $\frac{8}{25}$ .

$\frac{8^{2}}{25^{2}} {(4 - 5 (\frac{8}{25}))}^{3}$

Step 4.3.2.1.2

Raise $8$ to the power of $2$ .

$\frac{64}{25^{2}} {(4 - 5 (\frac{8}{25}))}^{3}$

Step 4.3.2.1.3

Raise $25$ to the power of $2$ .

$\frac{64}{625} {(4 - 5 (\frac{8}{25}))}^{3}$

Step 4.3.2.2

Simplify each term.

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Step 4.3.2.2.1

Cancel the common factor of $5$ .

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Step 4.3.2.2.1.1

Factor $5$ out of $- 5$ .

$\frac{64}{625} {(4 + 5 (- 1) \frac{8}{25})}^{3}$

Step 4.3.2.2.1.2

Factor $5$ out of $25$ .

$\frac{64}{625} {(4 + 5 \cdot - 1 \frac{8}{5 \cdot 5})}^{3}$

Step 4.3.2.2.1.3

Cancel the common factor.

$\frac{64}{625} {(4 + 5 \cdot - 1 \frac{8}{5 \cdot 5})}^{3}$

Step 4.3.2.2.1.4

Rewrite the expression.

$\frac{64}{625} {(4 - 1 (\frac{8}{5}))}^{3}$

Step 4.3.2.2.2

Rewrite $- 1 (\frac{8}{5})$ as $- (\frac{8}{5})$ .

$\frac{64}{625} {(4 - \frac{8}{5})}^{3}$

Step 4.3.2.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{64}{625} {(4 \cdot \frac{5}{5} - \frac{8}{5})}^{3}$

Step 4.3.2.4

Combine $4$ and $\frac{5}{5}$ .

$\frac{64}{625} {(\frac{4 \cdot 5}{5} - \frac{8}{5})}^{3}$

Step 4.3.2.5

Combine the numerators over the common denominator.

$\frac{64}{625} {(\frac{4 \cdot 5 - 8}{5})}^{3}$

Step 4.3.2.6

Simplify the numerator.

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Step 4.3.2.6.1

Multiply $4$ by $5$ .

$\frac{64}{625} {(\frac{20 - 8}{5})}^{3}$

Step 4.3.2.6.2

Subtract $8$ from $20$ .

$\frac{64}{625} {(\frac{12}{5})}^{3}$

Step 4.3.2.7

Combine fractions.

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Step 4.3.2.7.1

Apply the product rule to $\frac{12}{5}$ .

$\frac{64}{625} \cdot \frac{12^{3}}{5^{3}}$

Step 4.3.2.7.2

Combine.

$\frac{64 \cdot 12^{3}}{625 \cdot 5^{3}}$

Step 4.3.2.7.3

Raise $12$ to the power of $3$ .

$\frac{64 \cdot 1728}{625 \cdot 5^{3}}$

Step 4.3.2.8

Simplify the denominator.

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Step 4.3.2.8.1

Rewrite $625$ as $5^{4}$ .

$\frac{64 \cdot 1728}{5^{4} \cdot 5^{3}}$

Step 4.3.2.8.2

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

$\frac{64 \cdot 1728}{5^{4 + 3}}$

Step 4.3.2.8.3

Add $4$ and $3$ .

$\frac{64 \cdot 1728}{5^{7}}$

Step 4.3.2.9

Simplify the expression.

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Step 4.3.2.9.1

Multiply $64$ by $1728$ .

$\frac{110592}{5^{7}}$

Step 4.3.2.9.2

Raise $5$ to the power of $7$ .

$\frac{110592}{78125}$

Step 4.4

List all of the points.

$(0, 0), (\frac{4}{5}, 0), (\frac{8}{25}, \frac{110592}{78125})$

Step 5