Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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.

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

Step 3.1.2

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

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Step 3.1.2.1

Move $x$ .

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

Step 3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.1.3

Multiply $4$ by $4$ .

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

Step 3.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.1.5

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

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Step 3.1.5.1

Move $x^{2}$ .

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

Step 3.1.5.2

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

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Step 3.1.5.2.1

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

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

Step 3.1.5.2.2

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

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

Step 3.1.5.3

Add $2$ and $1$ .

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

Step 3.1.6

Multiply $4$ by $- 1$ .

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

Step 3.1.7

Rewrite using the commutative property of multiplication.

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

Step 3.1.8

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

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Step 3.1.8.1

Move $x$ .

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

Step 3.1.8.2

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

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Step 3.1.8.2.1

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

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

Step 3.1.8.2.2

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

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

Step 3.1.8.3

Add $1$ and $2$ .

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

Step 3.1.9

Multiply $- 1$ by $4$ .

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

Step 3.1.10

Rewrite using the commutative property of multiplication.

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

Step 3.1.11

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

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Step 3.1.11.1

Move $x^{2}$ .

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

Step 3.1.11.2

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

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

Step 3.1.11.3

Add $2$ and $2$ .

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

Step 3.1.12

Multiply $- 1$ by $- 1$ .

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

Step 3.1.13

Multiply $x^{4}$ by $1$ .

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

Step 3.2

Subtract $4 x^{3}$ from $- 4 x^{3}$ .

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