(x^2-3x)^2-8(x^2-3x)-20

2 min read Jun 17, 2024
(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).

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