Find the Domain V=x(6-2x)(6-2x)

$V = x (6 - 2 x) (6 - 2 x)$

Step 1

Simplify $x (6 - 2 x) (6 - 2 x)$ .

Simplify terms.

Apply the distributive property.

$V = (x \cdot 6 + x (- 2 x)) (6 - 2 x)$

Reorder.

Move $6$ to the left of $x$ .

$V = (6 \cdot x + x (- 2 x)) (6 - 2 x)$

Rewrite using the commutative property of multiplication.

$V = (6 \cdot x - 2 x \cdot x) (6 - 2 x)$

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

Move $x$ .

$V = (6 x - 2 (x \cdot x)) (6 - 2 x)$

Multiply $x$ by $x$ .

$V = (6 x - 2 x^{2}) (6 - 2 x)$

Expand $(6 x - 2 x^{2}) (6 - 2 x)$ using the FOIL Method.

Apply the distributive property.

$V = 6 x (6 - 2 x) - 2 x^{2} (6 - 2 x)$

Apply the distributive property.

$V = 6 x \cdot 6 + 6 x (- 2 x) - 2 x^{2} (6 - 2 x)$

Apply the distributive property.

$V = 6 x \cdot 6 + 6 x (- 2 x) - 2 x^{2} \cdot 6 - 2 x^{2} (- 2 x)$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

$V = 36 x + 6 x (- 2 x) - 2 x^{2} \cdot 6 - 2 x^{2} (- 2 x)$

Rewrite using the commutative property of multiplication.

$V = 36 x + 6 \cdot - 2 x \cdot x - 2 x^{2} \cdot 6 - 2 x^{2} (- 2 x)$

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

Move $x$ .

$V = 36 x + 6 \cdot - 2 (x \cdot x) - 2 x^{2} \cdot 6 - 2 x^{2} (- 2 x)$

Multiply $x$ by $x$ .

$V = 36 x + 6 \cdot - 2 x^{2} - 2 x^{2} \cdot 6 - 2 x^{2} (- 2 x)$

Multiply $6$ by $- 2$ .

$V = 36 x - 12 x^{2} - 2 x^{2} \cdot 6 - 2 x^{2} (- 2 x)$

Multiply $6$ by $- 2$ .

$V = 36 x - 12 x^{2} - 12 x^{2} - 2 x^{2} (- 2 x)$

Rewrite using the commutative property of multiplication.

$V = 36 x - 12 x^{2} - 12 x^{2} - 2 \cdot - 2 x^{2} x$

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

Move $x$ .

$V = 36 x - 12 x^{2} - 12 x^{2} - 2 \cdot - 2 (x \cdot x^{2})$

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

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

$V = 36 x - 12 x^{2} - 12 x^{2} - 2 \cdot - 2 (x^{1} x^{2})$

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

$V = 36 x - 12 x^{2} - 12 x^{2} - 2 \cdot - 2 x^{1 + 2}$

Add $1$ and $2$ .

$V = 36 x - 12 x^{2} - 12 x^{2} - 2 \cdot - 2 x^{3}$

Multiply $- 2$ by $- 2$ .

$V = 36 x - 12 x^{2} - 12 x^{2} + 4 x^{3}$

Subtract $12 x^{2}$ from $- 12 x^{2}$ .

$V = 36 x - 24 x^{2} + 4 x^{3}$

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

Find the Domain V=x(6-2x)(6-2x)

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