((x^2-2x-8)/(x^2-3x-10)) - ((x^2-2x-3)/(x^2+7x+6)) Please don't just give me the answer, I actually want to know how to do problems like this--the material the entire course was based on was taught using the learning style I cannot work with whatsoever and I retained none of it. This is more of a study question for me than anything. Please help.
I understand that you want to know how to solve the given problem rather than just getting the answer. Let's go step by step.
To simplify the expression:
((x^2-2x-8)/(x^2-3x-10)) - ((x^2-2x-3)/(x^2+7x+6))
First, let's factorize the denominators of both fractions to find their simplest form.
For the first fraction with the denominator (x^2-3x-10), we need to find two numbers whose sum is -3 and whose product is -10. The numbers are -5 and +2. Therefore, we can factorize the denominator as:
(x^2 - 3x - 10) = (x - 5)(x + 2)
For the second fraction with the denominator (x^2+7x+6), we need to find two numbers whose sum is 7 and whose product is 6. The numbers are 3 and 2. Therefore, we can factorize the denominator as:
(x^2 + 7x + 6) = (x + 3)(x + 2)
Now, we can rewrite the expression as:
[(x^2-2x-8)/((x - 5)(x + 2))] - [(x^2-2x-3)/((x + 3)(x + 2))]
Next, let's find the least common denominator (LCD) for both fractions. The LCD is the product of the common factors of the denominators, which are (x - 5), (x + 2), (x + 3).
Therefore, the LCD for the expression is (x - 5)(x + 2)(x + 3).
Now, we need to rewrite the fractions using the common denominator:
[(x^2-2x-8)/((x - 5)(x + 2))] - [(x^2-2x-3)/((x + 3)(x + 2))] =
[(x^2-2x-8)(x + 3)(x + 2) - (x^2-2x-3)(x - 5)] / [(x - 5)(x + 2)(x + 3)]
Next, we can simplify the numerator of the expression. Let's expand and simplify each bracket:
(x^2-2x-8)(x + 3)(x + 2) = [x(x^2+3x+2x+6) - 2(x^2+3x+2x+6) - 8(x + 3)(x + 2)]
Simplifying further:
= [x(x^2+5x+6) - 2(x^2+5x+6) - 8(x + 3)(x + 2)]
= [x^3+5x^2+6x - 2x^2-10x-12 - 8(x + 3)(x + 2)]
= [x^3+3x^2-4x-12 - 8(x^2+5x+6)]
Now let's expand and simplify the remaining bracket:
= [x^3+3x^2-4x-12 - 8x^2-40x-48]
= [x^3+3x^2-8x^2-4x-40x-12-48]
= [x^3-5x^2-44x-60]
Now, substitute this simplified numerator into the expression:
= [x^3-5x^2-44x-60] / [(x - 5)(x + 2)(x + 3)]
Thus, the simplified form of the original expression is (x^3-5x^2-44x-60) / ((x - 5)(x + 2)(x + 3)).
Please note that this process involves factoring, finding LCD, expanding, and simplifying the expression. It is crucial to pay attention to signs and perform the calculations carefully.
Please simplify this!! That's the question itself. Ugh, I'm so sorry.
see your earlier post, where I correctly guessed the proper expressions.
Rather than start a whole new thread, it would have been better to post your correction as part of the original question.