Find the LCD 8/(5x-15) , 14/(3x^2-18x+27)

$\frac{8}{5 x - 15}$ , $\frac{14}{3 x^{2} - 18 x + 27}$

Simplify each polynomial.

Factor $5$ out of $5 x - 15$ .

Factor $5$ out of $5 x$ .

$\frac{8}{5 (x) - 15}, \frac{14}{3 x^{2} - 18 x + 27}$

Factor $5$ out of $- 15$ .

$\frac{8}{5 x + 5 \cdot - 3}, \frac{14}{3 x^{2} - 18 x + 27}$

Factor $5$ out of $5 x + 5 \cdot - 3$ .

$\frac{8}{5 (x - 3)}, \frac{14}{3 x^{2} - 18 x + 27}$

Simplify the denominator.

Factor $3$ out of $3 x^{2} - 18 x + 27$ .

Factor $3$ out of $3 x^{2}$ .

$\frac{8}{5 (x - 3)}, \frac{14}{3 (x^{2}) - 18 x + 27}$

Factor $3$ out of $- 18 x$ .

$\frac{8}{5 (x - 3)}, \frac{14}{3 (x^{2}) + 3 (- 6 x) + 27}$

Factor $3$ out of $27$ .

$\frac{8}{5 (x - 3)}, \frac{14}{3 x^{2} + 3 (- 6 x) + 3 \cdot 9}$

Factor $3$ out of $3 x^{2} + 3 (- 6 x)$ .

$\frac{8}{5 (x - 3)}, \frac{14}{3 (x^{2} - 6 x) + 3 \cdot 9}$

Factor $3$ out of $3 (x^{2} - 6 x) + 3 \cdot 9$ .

$\frac{8}{5 (x - 3)}, \frac{14}{3 (x^{2} - 6 x + 9)}$

Factor using the perfect square rule.

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

$\frac{8}{5 (x - 3)}, \frac{14}{3 (x^{2} - 6 x + 3^{2})}$

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.

$\frac{8}{5 (x - 3)}, \frac{14}{3 (x^{2} - 2 \cdot x \cdot 3 + 3^{2})}$

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

$\frac{8}{5 (x - 3)}, \frac{14}{3 {(x - 3)}^{2}}$

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

$5 (x - 3), 3 {(x - 3)}^{2}$

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 $5$ has no factors besides $1$ and $5$ .

$5$ is a prime number

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

$3$ is a prime number

The LCM of $5, 3$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3 \cdot 5$

Multiply $3$ by $5$ .

$15$

The factor for $x - 3$ is $x - 3$ itself.

$(x - 3) = x - 3$

$(x - 3)$ occurs $1$ time.

The factors for $x - 3$ are $(x - 3) \cdot (x - 3)$ , which is $x - 3$ multiplied by itself $2$ times.

$(x - 3) = (x - 3) \cdot (x - 3)$

$(x - 3)$ occurs $2$ times.

The LCM of $x - 3, {(x - 3)}^{2}$ is the result of multiplying all factors the greatest number of times they occur in either term.

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

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

$15 {(x - 3)}^{2}$

Find the LCD 8/(5x-15) , 14/(3x^2-18x+27)

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