Algebra Examples

$2 a$ , $15 a^{3} + 18 a^{2} - 24 a$

Step 1

Factor $15 a^{3} + 18 a^{2} - 24 a$ .

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Step 1.1

Factor $3 a$ out of $15 a^{3} + 18 a^{2} - 24 a$ .

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Step 1.1.1

Factor $3 a$ out of $15 a^{3}$ .

$3 a (5 a^{2}) + 18 a^{2} - 24 a$

Step 1.1.2

Factor $3 a$ out of $18 a^{2}$ .

$3 a (5 a^{2}) + 3 a (6 a) - 24 a$

Step 1.1.3

Factor $3 a$ out of $- 24 a$ .

$3 a (5 a^{2}) + 3 a (6 a) + 3 a (- 8)$

Step 1.1.4

Factor $3 a$ out of $3 a (5 a^{2}) + 3 a (6 a)$ .

$3 a (5 a^{2} + 6 a) + 3 a (- 8)$

Step 1.1.5

Factor $3 a$ out of $3 a (5 a^{2} + 6 a) + 3 a (- 8)$ .

$3 a (5 a^{2} + 6 a - 8)$

Step 1.2

Factor.

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Step 1.2.1

Factor by grouping.

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Step 1.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 8 = - 40$ and whose sum is $b = 6$ .

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Step 1.2.1.1.1

Factor $6$ out of $6 a$ .

$3 a (5 a^{2} + 6 (a) - 8)$

Step 1.2.1.1.2

Rewrite $6$ as $- 4$ plus $10$

$3 a (5 a^{2} + (- 4 + 10) a - 8)$

Step 1.2.1.1.3

Apply the distributive property.

$3 a (5 a^{2} - 4 a + 10 a - 8)$

Step 1.2.1.2

Factor out the greatest common factor from each group.

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Step 1.2.1.2.1

Group the first two terms and the last two terms.

$3 a ((5 a^{2} - 4 a) + 10 a - 8)$

Step 1.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$3 a (a (5 a - 4) + 2 (5 a - 4))$

Step 1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $5 a - 4$ .

$3 a ((5 a - 4) (a + 2))$

Step 1.2.2

Remove unnecessary parentheses.

$3 a (5 a - 4) (a + 2)$

Step 2

Since $2 a, 3 a (5 a - 4) (a + 2)$ contains both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.

Steps to find the LCM for $2 a, 3 a (5 a - 4) (a + 2)$ are:

1. Find the LCM for the numeric part $2, 3$ .

2. Find the LCM for the variable part $a^{1}, a^{1}$ .

3. Find the LCM for the compound variable part $5 a - 4, a + 2$ .

4. Multiply each LCM together.

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

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

$2$ is a prime number

Step 5

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

$3$ is a prime number

Step 6

The LCM of $2, 3$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 3$

Step 7

Multiply $2$ by $3$ .

$6$

Step 8

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

$a^{1} = a$

$a$ occurs $1$ time.

Step 9

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

$a$

Step 10

The factor for $5 a - 4$ is $5 a - 4$ itself.

$(5 a - 4) = 5 a - 4$

$(5 a - 4)$ occurs $1$ time.

Step 11

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

$(a + 2) = a + 2$

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

Step 12

The LCM of $5 a - 4, a + 2$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(5 a - 4) (a + 2)$

Step 13

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

$6 a (5 a - 4) (a + 2)$