Find the LCM 22x^2+66x-220 , 4x^2-16

$22 x^{2} + 66 x - 220$ , $4 x^{2} - 16$

Step 1

Factor $22 x^{2} + 66 x - 220$ .

Factor $22$ out of $22 x^{2} + 66 x - 220$ .

Factor $22$ out of $22 x^{2}$ .

$22 (x^{2}) + 66 x - 220$

Factor $22$ out of $66 x$ .

$22 (x^{2}) + 22 (3 x) - 220$

Factor $22$ out of $- 220$ .

$22 x^{2} + 22 (3 x) + 22 \cdot - 10$

Factor $22$ out of $22 x^{2} + 22 (3 x)$ .

$22 (x^{2} + 3 x) + 22 \cdot - 10$

Factor $22$ out of $22 (x^{2} + 3 x) + 22 \cdot - 10$ .

$22 (x^{2} + 3 x - 10)$

Factor.

Factor $x^{2} + 3 x - 10$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 10$ and whose sum is $3$ .

$- 2, 5$

Write the factored form using these integers.

$22 ((x - 2) (x + 5))$

Remove unnecessary parentheses.

$22 (x - 2) (x + 5)$

Step 2

Factor $4 x^{2} - 16$ .

Factor $4$ out of $4 x^{2} - 16$ .

Factor $4$ out of $4 x^{2}$ .

$4 (x^{2}) - 16$

Factor $4$ out of $- 16$ .

$4 x^{2} + 4 \cdot - 4$

Factor $4$ out of $4 x^{2} + 4 \cdot - 4$ .

$4 (x^{2} - 4)$

Rewrite $4$ as $2^{2}$ .

$4 (x^{2} - 2^{2})$

Factor.

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

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

Remove unnecessary parentheses.

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

Step 3

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 4

$22$ has factors of $2$ and $11$ .

$2 \cdot 11$

Step 5

$4$ has factors of $2$ and $2$ .

$2 \cdot 2$

Step 6

The LCM of $22, 4$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2 \cdot 11$

Step 7

Multiply $2 \cdot 2 \cdot 11$ .

Multiply $2$ by $2$ .

$4 \cdot 11$

Multiply $4$ by $11$ .

$44$

Step 8

The factor for $x - 2$ is $x - 2$ itself.

$(x - 2) = x - 2$

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

Step 9

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

$(x + 5) = x + 5$

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

Step 10

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

$(x + 2) = x + 2$

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

Step 11

The factor for $x - 2$ is $x - 2$ itself.

$(x - 2) = x - 2$

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

Step 12

The LCM of $x - 2, x + 5, x + 2, x - 2$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x - 2) (x + 5) (x + 2)$

Step 13

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

$44 (x - 2) (x + 5) (x + 2)$

Find the LCM 22x^2+66x-220 , 4x^2-16

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