Find the LCM x^5-x , x^5-x^2 , x^5-x^3

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

Factor $x^{5} - x$ .

Factor $x$ out of $x^{5} - x$ .

Factor $x$ out of $x^{5}$ .

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

Factor $x$ out of $- x$ .

$x \cdot x^{4} + x \cdot - 1$

Factor $x$ out of $x \cdot x^{4} + x \cdot - 1$ .

$x (x^{4} - 1)$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x ({(x^{2})}^{2} - 1)$

Rewrite $1$ as $1^{2}$ .

$x ({(x^{2})}^{2} - 1^{2})$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 1$ .

$x ((x^{2} + 1) (x^{2} - 1))$

Factor.

Simplify.

Rewrite $1$ as $1^{2}$ .

$x ((x^{2} + 1) (x^{2} - 1^{2}))$

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 $b = 1$ .

$x ((x^{2} + 1) ((x + 1) (x - 1)))$

Remove unnecessary parentheses.

$x ((x^{2} + 1) (x + 1) (x - 1))$

Remove unnecessary parentheses.

$x (x^{2} + 1) (x + 1) (x - 1)$

Factor $x^{5} - x^{2}$ .

Factor $x^{2}$ out of $x^{5} - x^{2}$ .

Factor $x^{2}$ out of $x^{5}$ .

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

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

$x^{2} x^{3} + x^{2} \cdot - 1$

Factor $x^{2}$ out of $x^{2} x^{3} + x^{2} \cdot - 1$ .

$x^{2} (x^{3} - 1)$

Rewrite $1$ as $1^{3}$ .

$x^{2} (x^{3} - 1^{3})$

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

$x^{2} ((x - 1) (x^{2} + x \cdot 1 + 1^{2}))$

Factor.

Simplify.

Multiply $x$ by $1$ .

$x^{2} ((x - 1) (x^{2} + x + 1^{2}))$

One to any power is one.

$x^{2} ((x - 1) (x^{2} + x + 1))$

Remove unnecessary parentheses.

$x^{2} (x - 1) (x^{2} + x + 1)$

Factor $x^{5} - x^{3}$ .

Factor $x^{3}$ out of $x^{5} - x^{3}$ .

Factor $x^{3}$ out of $x^{5}$ .

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

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

$x^{3} x^{2} + x^{3} \cdot - 1$

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} \cdot - 1$ .

$x^{3} (x^{2} - 1)$

Rewrite $1$ as $1^{2}$ .

$x^{3} (x^{2} - 1^{2})$

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 $b = 1$ .

$x^{3} ((x + 1) (x - 1))$

Remove unnecessary parentheses.

$x^{3} (x + 1) (x - 1)$

Since $x (x^{2} + 1) (x + 1) (x - 1), x^{2} (x - 1) (x^{2} + x + 1), x^{3} (x + 1) (x - 1)$ contain 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 $x (x^{2} + 1) (x + 1) (x - 1), x^{2} (x - 1) (x^{2} + x + 1), x^{3} (x + 1) (x - 1)$ are:

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

2. Find the LCM for the variable part $x^{1}, x^{2}, x^{3}$ .

3. Find the LCM for the compound variable part $x^{2} + 1, x + 1, x - 1, x - 1, x^{2} + x + 1, x + 1, x - 1$ .

4. Multiply each LCM together.

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.

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

Not prime

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

$1$

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

$x^{1} = x$

$x$ occurs $1$ time.

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

The factors for $x^{3}$ are $x \cdot x \cdot x$ , which is $x$ multiplied by each other $3$ times.

$x^{3} = x \cdot x \cdot x$

$x$ occurs $3$ times.

The LCM of $x^{1}, x^{2}, x^{3}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x \cdot x$

Simplify $x \cdot x \cdot x$ .

Multiply $x$ by $x$ .

$x^{2} \cdot x$

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$x^{2} \cdot x^{1}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 + 1}$

Add $2$ and $1$ .

$x^{3}$

The factor for $x^{2} + 1$ is $x^{2} + 1$ itself.

$(x^{2} + 1) = x^{2} + 1$

$(x^{2} + 1)$ occurs $1$ time.

The factor for $x + 1$ is $x + 1$ itself.

$(x + 1) = x + 1$

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

The factor for $x - 1$ is $x - 1$ itself.

$(x - 1) = x - 1$

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

The factor for $x^{2} + x + 1$ is $x^{2} + x + 1$ itself.

$(x^{2} + x + 1) = x^{2} + x + 1$

$(x^{2} + x + 1)$ occurs $1$ time.

The factor for $x + 1$ is $x + 1$ itself.

$(x + 1) = x + 1$

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

The factor for $x - 1$ is $x - 1$ itself.

$(x - 1) = x - 1$

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

The LCM of $x^{2} + 1, x + 1, x - 1, x - 1, x^{2} + x + 1, x + 1, x - 1$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x^{2} + 1) (x + 1) (x - 1) (x^{2} + x + 1)$

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

$x^{3} (x^{2} + 1) (x + 1) (x - 1) (x^{2} + x + 1)$

Find the LCM x^5-x , x^5-x^2 , x^5-x^3

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