Find the LCM 3y^2-75 , y-5

$3 y^{2} - 75$ , $y - 5$

Step 1

Factor $3 y^{2} - 75$ .

Factor $3$ out of $3 y^{2} - 75$ .

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

$3 (y^{2}) - 75$

Factor $3$ out of $- 75$ .

$3 y^{2} + 3 \cdot - 25$

Factor $3$ out of $3 y^{2} + 3 \cdot - 25$ .

$3 (y^{2} - 25)$

Rewrite $25$ as $5^{2}$ .

$3 (y^{2} - 5^{2})$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 5$ .

$3 ((y + 5) (y - 5))$

Remove unnecessary parentheses.

$3 (y + 5) (y - 5)$

Step 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.

Step 3

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

$3$ is a prime number

Step 4

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

Not prime

Step 5

The LCM of $3, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3$

Step 6

The factor for $y + 5$ is $y + 5$ itself.

$(y + 5) = y + 5$

$(y + 5)$ occurs $1$ time.

Step 7

The factor for $y - 5$ is $y - 5$ itself.

$(y - 5) = y - 5$

$(y - 5)$ occurs $1$ time.

Step 8

The LCM of $y + 5, y - 5, y - 5$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(y + 5) (y - 5)$

Step 9

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

$3 (y + 5) (y - 5)$