How do I factor this:
x^6-124x^3-125
well, I would let z = x^3
then
z^2 -124 z -125
(z-125)(z+1)
so
(x^3-125)(x^3 + 1)
we know 5^3 = 125
try dividing by (x^3-125) by x-5
sure enough
(x-5)(x^2+5x +25)(x^3 +1)
so try (x+1) on the (x^3+1) part
and get
(x-5)(x^2+5x+25)(x+1)(x^2-x+1)
To factor the expression x^6 - 124x^3 - 125, we can use a technique called factoring by grouping. Here's how you can do it step by step:
Step 1: Recognize the form
The given expression is in the form of a difference of cubes, which can be factored using a specific formula. The formula for a difference of cubes is:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Step 2: Rewrite the expression
Rewrite the expression as:
x^6 - 5x^3 + x^3 - 125
Step 3: Group the terms
Group the terms into two pairs:
(x^6 - 5x^3) + (x^3 - 125)
Step 4: Factor out common factors from each group
From the first group, factor out x^3:
x^3(x^3 - 5)
From the second group, recognize it as a difference of cubes:
(x - 5)(x^2 + 5x + 25)
Step 5: Combining the factors
Now, we can combine the factors from both groups:
x^3(x^3 - 5) + (x - 5)(x^2 + 5x + 25)
Factoring is now complete. The fully factored expression is:
x^3(x^3 - 5) + (x - 5)(x^2 + 5x + 25)