Calculus Examples

{(4 x - x^{2})}^{2}

${(4 x - x^{2})}^{2}$

Step 1

Rewrite

{(4 x - x^{2})}^{2}

${(4 x - x^{2})}^{2}$ as

(4 x - x^{2}) (4 x - x^{2})

$(4 x - x^{2}) (4 x - x^{2})$ .

(4 x - x^{2}) (4 x - x^{2})

$(4 x - x^{2}) (4 x - x^{2})$

Step 2

Expand

(4 x - x^{2}) (4 x - x^{2})

$(4 x - x^{2}) (4 x - x^{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

4 x (4 x - x^{2}) - x^{2} (4 x - x^{2})

$4 x (4 x - x^{2}) - x^{2} (4 x - x^{2})$

Step 2.2

Apply the distributive property.

4 x (4 x) + 4 x (- x^{2}) - x^{2} (4 x - x^{2})

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

Step 2.3

Apply the distributive property.

4 x (4 x) + 4 x (- x^{2}) - x^{2} (4 x) - x^{2} (- x^{2})

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

4 x (4 x) + 4 x (- x^{2}) - x^{2} (4 x) - x^{2} (- x^{2})

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

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

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

Step 3.1.2

Multiply

x

$x$ by

x

$x$ by adding the exponents.

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Step 3.1.2.1

Move

x

$x$ .

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

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

Step 3.1.2.2

Multiply

x

$x$ by

x

$x$ .

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

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

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

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

Step 3.1.3

Multiply

4

$4$ by

4

$4$ .

16 x^{2} + 4 x (- x^{2}) - x^{2} (4 x) - x^{2} (- x^{2})

$16 x^{2} + 4 x (- x^{2}) - x^{2} (4 x) - x^{2} (- x^{2})$

Step 3.1.4

Rewrite using the commutative property of multiplication.

16 x^{2} + 4 \cdot - 1 x \cdot x^{2} - x^{2} (4 x) - x^{2} (- x^{2})

$16 x^{2} + 4 \cdot - 1 x \cdot x^{2} - x^{2} (4 x) - x^{2} (- x^{2})$

Step 3.1.5

Multiply

x

$x$ by

x^{2}

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

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Step 3.1.5.1

Move

x^{2}

$x^{2}$ .

16 x^{2} + 4 \cdot - 1 (x^{2} x) - x^{2} (4 x) - x^{2} (- x^{2})

$16 x^{2} + 4 \cdot - 1 (x^{2} x) - x^{2} (4 x) - x^{2} (- x^{2})$

Step 3.1.5.2

Multiply

x^{2}

$x^{2}$ by

x

$x$ .

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Step 3.1.5.2.1

Raise

x

$x$ to the power of

1

$1$ .

16 x^{2} + 4 \cdot - 1 (x^{2} x^{1}) - x^{2} (4 x) - x^{2} (- x^{2})

$16 x^{2} + 4 \cdot - 1 (x^{2} x^{1}) - x^{2} (4 x) - x^{2} (- x^{2})$

Step 3.1.5.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

16 x^{2} + 4 \cdot - 1 x^{2 + 1} - x^{2} (4 x) - x^{2} (- x^{2})

$16 x^{2} + 4 \cdot - 1 x^{2 + 1} - x^{2} (4 x) - x^{2} (- x^{2})$

16 x^{2} + 4 \cdot - 1 x^{2 + 1} - x^{2} (4 x) - x^{2} (- x^{2})

$16 x^{2} + 4 \cdot - 1 x^{2 + 1} - x^{2} (4 x) - x^{2} (- x^{2})$

Step 3.1.5.3

Add

2

$2$ and

1

$1$ .

16 x^{2} + 4 \cdot - 1 x^{3} - x^{2} (4 x) - x^{2} (- x^{2})

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

16 x^{2} + 4 \cdot - 1 x^{3} - x^{2} (4 x) - x^{2} (- x^{2})

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

Step 3.1.6

Multiply

4

$4$ by

- 1

$- 1$ .

16 x^{2} - 4 x^{3} - x^{2} (4 x) - x^{2} (- x^{2})

$16 x^{2} - 4 x^{3} - x^{2} (4 x) - x^{2} (- x^{2})$

Step 3.1.7

Rewrite using the commutative property of multiplication.

16 x^{2} - 4 x^{3} - 1 \cdot 4 x^{2} x - x^{2} (- x^{2})

$16 x^{2} - 4 x^{3} - 1 \cdot 4 x^{2} x - x^{2} (- x^{2})$

Step 3.1.8

Multiply

x^{2}

$x^{2}$ by

x

$x$ by adding the exponents.

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Step 3.1.8.1

Move

x

$x$ .

16 x^{2} - 4 x^{3} - 1 \cdot 4 (x \cdot x^{2}) - x^{2} (- x^{2})

$16 x^{2} - 4 x^{3} - 1 \cdot 4 (x \cdot x^{2}) - x^{2} (- x^{2})$

Step 3.1.8.2

Multiply

x

$x$ by

x^{2}

$x^{2}$ .

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Step 3.1.8.2.1

Raise

x

$x$ to the power of

1

$1$ .

16 x^{2} - 4 x^{3} - 1 \cdot 4 (x^{1} x^{2}) - x^{2} (- x^{2})

$16 x^{2} - 4 x^{3} - 1 \cdot 4 (x^{1} x^{2}) - x^{2} (- x^{2})$

Step 3.1.8.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

16 x^{2} - 4 x^{3} - 1 \cdot 4 x^{1 + 2} - x^{2} (- x^{2})

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

16 x^{2} - 4 x^{3} - 1 \cdot 4 x^{1 + 2} - x^{2} (- x^{2})

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

Step 3.1.8.3

Add

1

$1$ and

2

$2$ .

16 x^{2} - 4 x^{3} - 1 \cdot 4 x^{3} - x^{2} (- x^{2})

$16 x^{2} - 4 x^{3} - 1 \cdot 4 x^{3} - x^{2} (- x^{2})$

16 x^{2} - 4 x^{3} - 1 \cdot 4 x^{3} - x^{2} (- x^{2})

$16 x^{2} - 4 x^{3} - 1 \cdot 4 x^{3} - x^{2} (- x^{2})$

Step 3.1.9

Multiply

- 1

$- 1$ by

4

$4$ .

16 x^{2} - 4 x^{3} - 4 x^{3} - x^{2} (- x^{2})

$16 x^{2} - 4 x^{3} - 4 x^{3} - x^{2} (- x^{2})$

Step 3.1.10

Rewrite using the commutative property of multiplication.

16 x^{2} - 4 x^{3} - 4 x^{3} - 1 \cdot - 1 x^{2} x^{2}

$16 x^{2} - 4 x^{3} - 4 x^{3} - 1 \cdot - 1 x^{2} x^{2}$

Step 3.1.11

Multiply

x^{2}

$x^{2}$ by

x^{2}

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

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Step 3.1.11.1

Move

x^{2}

$x^{2}$ .

16 x^{2} - 4 x^{3} - 4 x^{3} - 1 \cdot - 1 (x^{2} x^{2})

$16 x^{2} - 4 x^{3} - 4 x^{3} - 1 \cdot - 1 (x^{2} x^{2})$

Step 3.1.11.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

16 x^{2} - 4 x^{3} - 4 x^{3} - 1 \cdot - 1 x^{2 + 2}

$16 x^{2} - 4 x^{3} - 4 x^{3} - 1 \cdot - 1 x^{2 + 2}$

Step 3.1.11.3

Add

2

$2$ and

2

$2$ .

16 x^{2} - 4 x^{3} - 4 x^{3} - 1 \cdot - 1 x^{4}

$16 x^{2} - 4 x^{3} - 4 x^{3} - 1 \cdot - 1 x^{4}$

16 x^{2} - 4 x^{3} - 4 x^{3} - 1 \cdot - 1 x^{4}

$16 x^{2} - 4 x^{3} - 4 x^{3} - 1 \cdot - 1 x^{4}$

Step 3.1.12

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

16 x^{2} - 4 x^{3} - 4 x^{3} + 1 x^{4}

$16 x^{2} - 4 x^{3} - 4 x^{3} + 1 x^{4}$

Step 3.1.13

Multiply

x^{4}

$x^{4}$ by

1

$1$ .

16 x^{2} - 4 x^{3} - 4 x^{3} + x^{4}

$16 x^{2} - 4 x^{3} - 4 x^{3} + x^{4}$

16 x^{2} - 4 x^{3} - 4 x^{3} + x^{4}

$16 x^{2} - 4 x^{3} - 4 x^{3} + x^{4}$

Step 3.2

Subtract

4 x^{3}

$4 x^{3}$ from

- 4 x^{3}

$- 4 x^{3}$ .

16 x^{2} - 8 x^{3} + x^{4}

$16 x^{2} - 8 x^{3} + x^{4}$

16 x^{2} - 8 x^{3} + x^{4}

$16 x^{2} - 8 x^{3} + x^{4}$