Find the LCD (9-x^2)/(3x) and (x^2+6x+9)/(3x)

$\frac{9 - x^{2}}{3 x}$ and $\frac{x^{2} + 6 x + 9}{3 x}$

Step 1

Simplify each polynomial.

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

$\frac{3^{2} - x^{2}}{3 x}, \frac{x^{2} + 6 x + 9}{3 x}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

$\frac{(3 + x) (3 - x)}{3 x}, \frac{x^{2} + 6 x + 9}{3 x}$

Factor using the perfect square rule.

Rewrite $9$ as $3^{2}$ .

$\frac{(3 + x) (3 - x)}{3 x}, \frac{x^{2} + 6 x + 3^{2}}{3 x}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Rewrite the polynomial.

$\frac{(3 + x) (3 - x)}{3 x}, \frac{x^{2} + 2 \cdot x \cdot 3 + 3^{2}}{3 x}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

$\frac{(3 + x) (3 - x)}{3 x}, \frac{{(x + 3)}^{2}}{3 x}$

Step 2

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

$3 x, 3 x$

Step 3

Since $3 x, 3 x$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $3, 3$ then find LCM for the variable part $x^{1}, x^{1}$ .

Step 4

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.

Step 5

Since $3$ has no factors besides $1$ and $3$ .

$3$ is a prime number

Step 6

The LCM of $3, 3$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3$

Step 7

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

Step 8

The LCM of $x^{1}, x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x$

Step 9

The LCM for $3 x, 3 x$ is the numeric part $3$ multiplied by the variable part.

$3 x$

Find the LCD (9-x^2)/(3x) and (x^2+6x+9)/(3x)

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