Please simplify the difference!: ((x^2+2x+8)/(x^2-3x-10)) - ((x^2-2x-3)/(x^2+7x-6)). The material the entire course was based on was taught using a learning style I cannot work with whatsoever and because of that, I retained nothing from the course. This is more of a study question for me than anything. Please help...
If the question is wrong, how do you expect help?
Anyway, working with what you gave,
(x^2+2x+8)/(x^2-3x-10) - (x^2-2x-3)/(x^2+7x-6)
= (x^2+2x+8)/((x-5)(x+2)) - (x+1)(x-3)/(x^2+7x-6)
I'm sorry, but that does not simplify too well. Let's try
(x^2-2x-8)/(x^2-3x-10) - (x^2-2x-3)/(x^2+7x+6)
= (x-4)(x+2) / (x-5)(x+2) - (x+1)(x-3) / (x+1)(x+6)
= (x-4)/(x-5) - (x-3)/(x+6)
= ((x-4)(x+6) - (x-3)(x-5)) / ((x-5)(x-6))
= (10x-39)/(x^2-11x+30)
To simplify the given expression, we need to find a common denominator for both fractions and combine them. Here are the steps to simplify the difference:
1. Factorize the denominators:
The first denominator, x^2 - 3x - 10, can be factored as (x - 5)(x + 2).
The second denominator, x^2 + 7x - 6, can be factored as (x - 1)(x + 6).
2. Find the least common denominator (LCD):
The LCD is the product of the distinct factors from both denominators, so it is (x - 5)(x + 2)(x - 1)(x + 6).
3. Rewrite the fractions with the common denominator:
The first fraction becomes ((x^2 + 2x + 8)(x - 1)(x + 6))/[(x - 5)(x + 2)(x - 1)(x + 6)].
The second fraction becomes ((x^2 - 2x - 3)(x - 5)(x + 2))/[(x - 5)(x + 2)(x - 1)(x + 6)].
4. Combine the fractions:
Subtract the second fraction from the first:
((x^2 + 2x + 8)(x - 1)(x + 6) - (x^2 - 2x - 3)(x - 5)(x + 2))/[(x - 5)(x + 2)(x - 1)(x + 6)].
5. Expand and simplify:
Multiply out each expression numerically and combine like terms:
(x^3 - 13x^2 - 70x + 56 - x^3 + 8x^2 + 25x - 24)/[(x - 5)(x + 2)(x - 1)(x + 6)].
6. Simplify further:
Combine like terms in the numerator:
(-5x^2 - 45x + 32)/[(x - 5)(x + 2)(x - 1)(x + 6)].
So, the simplified form of the given expression is (-5x^2 - 45x + 32)/[(x - 5)(x + 2)(x - 1)(x + 6)].