Find the LCD 3/((x-2)+x) , (3x)/(4x-(x-4)) , (6 square root of 2)/(x-(3x+1))

$\frac{3}{(x - 2) + x}$ , $\frac{3 x}{4 x - (x - 4)}$ , $\frac{6 \sqrt{2}}{x - (3 x + 1)}$

Simplify each polynomial.

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Simplify the denominator.

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Add $x$ and $x$ .

$\frac{3}{2 x - 2}, \frac{3 x}{4 x - (x - 4)}, \frac{6 \sqrt{2}}{x - (3 x + 1)}$

Factor $2$ out of $2 x - 2$ .

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Factor $2$ out of $2 x$ .

$\frac{3}{2 (x) - 2}, \frac{3 x}{4 x - (x - 4)}, \frac{6 \sqrt{2}}{x - (3 x + 1)}$

Factor $2$ out of $- 2$ .

$\frac{3}{2 (x) + 2 (- 1)}, \frac{3 x}{4 x - (x - 4)}, \frac{6 \sqrt{2}}{x - (3 x + 1)}$

Factor $2$ out of $2 (x) + 2 (- 1)$ .

$\frac{3}{2 (x - 1)}, \frac{3 x}{4 x - (x - 4)}, \frac{6 \sqrt{2}}{x - (3 x + 1)}$

Simplify the denominator.

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Apply the distributive property.

$\frac{3}{2 (x - 1)}, \frac{3 x}{4 x - x + 4}, \frac{6 \sqrt{2}}{x - (3 x + 1)}$

Multiply $- 1$ by $- 4$ .

$\frac{3}{2 (x - 1)}, \frac{3 x}{4 x - x + 4}, \frac{6 \sqrt{2}}{x - (3 x + 1)}$

Subtract $x$ from $4 x$ .

$\frac{3}{2 (x - 1)}, \frac{3 x}{3 x + 4}, \frac{6 \sqrt{2}}{x - (3 x + 1)}$

Simplify the denominator.

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Apply the distributive property.

$\frac{3}{2 (x - 1)}, \frac{3 x}{3 x + 4}, \frac{6 \sqrt{2}}{x - (3 x) - 1 \cdot 1}$

Multiply $3$ by $- 1$ .

$\frac{3}{2 (x - 1)}, \frac{3 x}{3 x + 4}, \frac{6 \sqrt{2}}{x - 3 x - 1 \cdot 1}$

Multiply $- 1$ by $1$ .

$\frac{3}{2 (x - 1)}, \frac{3 x}{3 x + 4}, \frac{6 \sqrt{2}}{x - 3 x - 1}$

Subtract $3 x$ from $x$ .

$\frac{3}{2 (x - 1)}, \frac{3 x}{3 x + 4}, \frac{6 \sqrt{2}}{- 2 x - 1}$

Simplify with factoring out.

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Factor $- 1$ out of $- 2 x$ .

$\frac{3}{2 (x - 1)}, \frac{3 x}{3 x + 4}, \frac{6 \sqrt{2}}{- (2 x) - 1}$

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{3}{2 (x - 1)}, \frac{3 x}{3 x + 4}, \frac{6 \sqrt{2}}{- (2 x) - 1 \cdot 1}$

Factor $- 1$ out of $- (2 x) - 1 (1)$ .

$\frac{3}{2 (x - 1)}, \frac{3 x}{3 x + 4}, \frac{6 \sqrt{2}}{- (2 x + 1)}$

Rewrite negatives.

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Rewrite $- (2 x + 1)$ as $- 1 (2 x + 1)$ .

$\frac{3}{2 (x - 1)}, \frac{3 x}{3 x + 4}, \frac{6 \sqrt{2}}{- 1 (2 x + 1)}$

Move the negative in front of the fraction.

$\frac{3}{2 (x - 1)}, \frac{3 x}{3 x + 4}, - \frac{6 \sqrt{2}}{2 x + 1}$

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 (x - 1), 3 x + 4, 2 x + 1$

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $2, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

The factor for $x - 1$ is $x - 1$ itself.

$(x - 1) = x - 1$

$(x - 1)$ occurs $1$ time.

The factor for $3 x + 4$ is $3 x + 4$ itself.

$(3 x + 4) = 3 x + 4$

$(3 x + 4)$ occurs $1$ time.

The factor for $2 x + 1$ is $2 x + 1$ itself.

$(2 x + 1) = 2 x + 1$

$(2 x + 1)$ occurs $1$ time.

The LCM of $x - 1, 3 x + 4, 2 x + 1$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x - 1) (3 x + 4) (2 x + 1)$

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$2 (x - 1) (3 x + 4) (2 x + 1)$

Find the LCD 3/((x-2)+x) , (3x)/(4x-(x-4)) , (6 square root of 2)/(x-(3x+1))