Basic Math Examples

$c^{6} - 133 c^{3} + 1000$

Step 1

Rewrite $c^{6}$ as ${(c^{3})}^{2}$ .

${(c^{3})}^{2} - 133 c^{3} + 1000$

Step 2

Let $u = c^{3}$ . Substitute $u$ for all occurrences of $c^{3}$ .

$u^{2} - 133 u + 1000$

Step 3

Factor $u^{2} - 133 u + 1000$ using the AC method.

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Step 3.1

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 $1000$ and whose sum is $- 133$ .

$- 125, - 8$

Step 3.2

Write the factored form using these integers.

$(u - 125) (u - 8)$

Step 4

Replace all occurrences of $u$ with $c^{3}$ .

$(c^{3} - 125) (c^{3} - 8)$

Step 5

Rewrite $125$ as $5^{3}$ .

$(c^{3} - 5^{3}) (c^{3} - 8)$

Step 6

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 $a = c$ and $b = 5$ .

$(c - 5) (c^{2} + c \cdot 5 + 5^{2}) (c^{3} - 8)$

Step 7

Simplify.

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Step 7.1

Move $5$ to the left of $c$ .

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

Step 7.2

Raise $5$ to the power of $2$ .

$(c - 5) (c^{2} + 5 c + 25) (c^{3} - 8)$

Step 8

Rewrite $8$ as $2^{3}$ .

$(c - 5) (c^{2} + 5 c + 25) (c^{3} - 2^{3})$

Step 9

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 $a = c$ and $b = 2$ .

$(c - 5) (c^{2} + 5 c + 25) ((c - 2) (c^{2} + c \cdot 2 + 2^{2}))$

Step 10

Factor.

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Step 10.1

Simplify.

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Step 10.1.1

Move $2$ to the left of $c$ .

$(c - 5) (c^{2} + 5 c + 25) ((c - 2) (c^{2} + 2 c + 2^{2}))$

Step 10.1.2

Raise $2$ to the power of $2$ .

$(c - 5) (c^{2} + 5 c + 25) ((c - 2) (c^{2} + 2 c + 4))$

Step 10.2

Remove unnecessary parentheses.

$(c - 5) (c^{2} + 5 c + 25) (c - 2) (c^{2} + 2 c + 4)$