Simplify the following expression completely.
(x^2 - x - 6)(x^2 - 2x - 3)
To simplify the expression, we can use the distributive property to multiply each term in the first expression by each term in the second expression.
(x^2 - x - 6)(x^2 - 2x - 3)
= x^2(x^2 - 2x - 3) - x(x^2 - 2x - 3) - 6(x^2 - 2x - 3)
= x^4 - 2x^3 - 3x^2 - x^3 + 2x^2 + 3x - 6x^2 + 12x + 18
= x^4 - 3x^3 - 7x^2 + 15x + 18
Therefore, the completely simplified expression is x^4 - 3x^3 - 7x^2 + 15x + 18.
Answer must be in (x - ?)/(x - ?)
To simplify the expression further, we can factor the expression:
x^4 - 3x^3 - 7x^2 + 15x + 18
Grouping the terms, we have:
(x^4 - 3x^3) - (7x^2 - 15x - 18)
Now, let's factor out common factors from each group:
x^3(x - 3) - 3(7x^2 - 15x - 18)
Next, let's factor the quadratic term inside the parentheses:
7x^2 - 15x - 18 = (7x + 6)(x - 3)
Replacing this back into our expression:
x^3(x - 3) - 3(7x + 6)(x - 3)
We notice that (x - 3) appears in both terms, so we can factor it out:
(x - 3)(x^3 - 3(7x + 6))
Expanding the second factor, we have:
(x - 3)(x^3 - 21x - 18)
Therefore, the completely simplified expression is:
(x - 3)(x^3 - 21x - 18).
Answer must be in binomial factor
My apologies for the misunderstanding. Let's try again.
(x^2 - x - 6)(x^2 - 2x - 3)
To simplify this expression, we can factor each quadratic individually:
(x^2 - x - 6) factors to (x - 3)(x + 2)
(x^2 - 2x - 3) factors to (x - 3)(x + 1)
Now, we can rewrite the expression using these factors:
(x - 3)(x + 2)(x - 3)(x + 1)
We can further simplify this by combining like terms:
(x - 3)(x - 3)(x + 2)(x + 1)
Now, we can write it in the form (x - ?)/(x - ?):
(x - 3)(x - 3)(x + 2)(x + 1) = (x - 3)^2(x + 2)(x + 1)
So, the completely simplified expression is (x - 3)^2(x + 2)(x + 1).