c6-133c3+1000

Rewrite c6 as (c3)2.

(c3)2-133c3+1000

Let u=c3. Substitute u for all occurrences of c3.

u2-133u+1000

Consider the form x2+bx+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

Write the factored form using these integers.

(u-125)(u-8)

(u-125)(u-8)

Replace all occurrences of u with c3.

(c3-125)(c3-8)

Rewrite 125 as 53.

(c3-53)(c3-8)

Since both terms are perfect cubes, factor using the difference of cubes formula, a3-b3=(a-b)(a2+ab+b2) where a=c and b=5.

(c-5)(c2+c⋅5+52)(c3-8)

Move 5 to the left of c.

(c-5)(c2+5c+52)(c3-8)

Raise 5 to the power of 2.

(c-5)(c2+5c+25)(c3-8)

(c-5)(c2+5c+25)(c3-8)

Rewrite 8 as 23.

(c-5)(c2+5c+25)(c3-23)

Since both terms are perfect cubes, factor using the difference of cubes formula, a3-b3=(a-b)(a2+ab+b2) where a=c and b=2.

(c-5)(c2+5c+25)((c-2)(c2+c⋅2+22))

Simplify.

Move 2 to the left of c.

(c-5)(c2+5c+25)((c-2)(c2+2c+22))

Raise 2 to the power of 2.

(c-5)(c2+5c+25)((c-2)(c2+2c+4))

(c-5)(c2+5c+25)((c-2)(c2+2c+4))

Remove unnecessary parentheses.

(c-5)(c2+5c+25)(c-2)(c2+2c+4)

(c-5)(c2+5c+25)(c-2)(c2+2c+4)

Factor c^6-133c^3+1000