Linear Algebra Examples

$V = x (14 - 2 x) (14 - 2 x)$

Step 1

Simplify $x (14 - 2 x) (14 - 2 x)$ .

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Step 1.1

Simplify terms.

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Step 1.1.1

Apply the distributive property.

$V = (x \cdot 14 + x (- 2 x)) (14 - 2 x)$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Move $14$ to the left of $x$ .

$V = (14 \cdot x + x (- 2 x)) (14 - 2 x)$

Step 1.1.2.2

Rewrite using the commutative property of multiplication.

$V = (14 \cdot x - 2 x \cdot x) (14 - 2 x)$

Step 1.1.3

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

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Step 1.1.3.1

Move $x$ .

$V = (14 x - 2 (x \cdot x)) (14 - 2 x)$

Step 1.1.3.2

Multiply $x$ by $x$ .

$V = (14 x - 2 x^{2}) (14 - 2 x)$

Step 1.2

Expand $(14 x - 2 x^{2}) (14 - 2 x)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$V = 14 x (14 - 2 x) - 2 x^{2} (14 - 2 x)$

Step 1.2.2

Apply the distributive property.

$V = 14 x \cdot 14 + 14 x (- 2 x) - 2 x^{2} (14 - 2 x)$

Step 1.2.3

Apply the distributive property.

$V = 14 x \cdot 14 + 14 x (- 2 x) - 2 x^{2} \cdot 14 - 2 x^{2} (- 2 x)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $14$ by $14$ .

$V = 196 x + 14 x (- 2 x) - 2 x^{2} \cdot 14 - 2 x^{2} (- 2 x)$

Step 1.3.1.2

Rewrite using the commutative property of multiplication.

$V = 196 x + 14 \cdot - 2 x \cdot x - 2 x^{2} \cdot 14 - 2 x^{2} (- 2 x)$

Step 1.3.1.3

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

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Step 1.3.1.3.1

Move $x$ .

$V = 196 x + 14 \cdot - 2 (x \cdot x) - 2 x^{2} \cdot 14 - 2 x^{2} (- 2 x)$

Step 1.3.1.3.2

Multiply $x$ by $x$ .

$V = 196 x + 14 \cdot - 2 x^{2} - 2 x^{2} \cdot 14 - 2 x^{2} (- 2 x)$

Step 1.3.1.4

Multiply $14$ by $- 2$ .

$V = 196 x - 28 x^{2} - 2 x^{2} \cdot 14 - 2 x^{2} (- 2 x)$

Step 1.3.1.5

Multiply $14$ by $- 2$ .

$V = 196 x - 28 x^{2} - 28 x^{2} - 2 x^{2} (- 2 x)$

Step 1.3.1.6

Rewrite using the commutative property of multiplication.

$V = 196 x - 28 x^{2} - 28 x^{2} - 2 \cdot - 2 x^{2} x$

Step 1.3.1.7

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

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Step 1.3.1.7.1

Move $x$ .

$V = 196 x - 28 x^{2} - 28 x^{2} - 2 \cdot - 2 (x \cdot x^{2})$

Step 1.3.1.7.2

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

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Step 1.3.1.7.2.1

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

$V = 196 x - 28 x^{2} - 28 x^{2} - 2 \cdot - 2 (x^{1} x^{2})$

Step 1.3.1.7.2.2

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

$V = 196 x - 28 x^{2} - 28 x^{2} - 2 \cdot - 2 x^{1 + 2}$

Step 1.3.1.7.3

Add $1$ and $2$ .

$V = 196 x - 28 x^{2} - 28 x^{2} - 2 \cdot - 2 x^{3}$

Step 1.3.1.8

Multiply $- 2$ by $- 2$ .

$V = 196 x - 28 x^{2} - 28 x^{2} + 4 x^{3}$

Step 1.3.2

Subtract $28 x^{2}$ from $- 28 x^{2}$ .

$V = 196 x - 56 x^{2} + 4 x^{3}$

Step 2

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3