factor.
x^6 - 124 x^3 - 125
x^7 - 8x^4 - 16x^3 + 128
first one:
x^6 - 124 x^3 - 125
= (x^3 + 1)(x^3 - 125)
now each of those factors as the sum/difference of cubes to get
(x+1)(x^2 - x + 1)(x+5)(x^2 - 5x + 25)
the second is a "grouping" kind at the start ...
x^7 - 8x^4 - 16x^3 + 128
= x^4(x^3 - 8) - 16(x^3 - 8)
= (x^3 - 8)(x^4 - 16)
= (x-2)(x^2 + 2x + 4)(x^2 + 4)(x+2)(x-2)
To factor the expressions x^6 - 124x^3 - 125 and x^7 - 8x^4 - 16x^3 + 128, we can use the factoring techniques.
Let's start with the expression x^6 - 124x^3 - 125.
Step 1: Notice that both terms have a common factor of x^3. We can factor it out:
x^6 - 124x^3 - 125 = x^3(x^3 - 124) - 125
Step 2: The expression x^3 - 124 cannot be factored any further, so now we can focus on the terms (x^3 - 124) - 125. Notice that this is a difference of cubes expression.
Step 3: Now, let's use the formula for the difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2).
In our case, a = x and b = 5. So we have:
(x^3 - 5^3) = (x - 5)(x^2 + 5x + 25)
Combining these steps, the factored form of x^6 - 124x^3 - 125 is:
x^3(x - 5)(x^2 + 5x + 25).
Now let's move on to the expression x^7 - 8x^4 - 16x^3 + 128.
Step 1: Notice that all terms have a common factor of x^3. We can factor it out:
x^7 - 8x^4 - 16x^3 + 128 = x^3(x^4 - 8x - 16) + 128
Step 2: The expression x^4 - 8x - 16 cannot be factored any further, so now we can focus on the terms (x^4 - 8x - 16) + 128.
Step 3: There is no specific formula to factor a quartic equation, so we need to look for possible factors or simplify the expression.
Since we can't factor it any further, x^4 - 8x - 16 cannot be factored.
Combining these steps, the factored form of x^7 - 8x^4 - 16x^3 + 128 is:
x^3(x^4 - 8x - 16) + 128.
Please note that factoring expressions can sometimes be tricky, and there might not always be a simple factorization. In such cases, the expressions may remain in their factored or partially factored forms.