%5E2-8(x%5E2-3x)-20.jpeg "(x^2-3x)^2-8(x^2-3x)-20")
Factoring the Expression: (x^2 - 3x)^2 - 8(x^2 - 3x) - 20
This expression might look intimidating at first, but we can simplify it by using a clever substitution and applying factoring techniques.
1. Substitution
Let's substitute u = x^2 - 3x. This will make the expression much easier to work with:
(x^2 - 3x)^2 - 8(x^2 - 3x) - 20 becomes u^2 - 8u - 20
2. Factoring the Quadratic
Now we have a simple quadratic expression. We need to find two numbers that multiply to -20 and add up to -8. These numbers are -10 and 2:
u^2 - 8u - 20 = (u - 10)(u + 2)
3. Substitute Back
Remember, we substituted u for x^2 - 3x. Let's substitute it back in:
(u - 10)(u + 2) = (x^2 - 3x - 10)(x^2 - 3x + 2)
4. Factoring Further
We can factor the two remaining expressions:
(x^2 - 3x - 10)(x^2 - 3x + 2) = (x - 5)(x + 2)(x - 1)(x - 2)
Final Result
Therefore, the factored form of (x^2 - 3x)^2 - 8(x^2 - 3x) - 20 is (x - 5)(x + 2)(x - 1)(x - 2).