simplify and state any restrictions:
1. x+3/x^2+11x+24 - 2x+10/x^2+11x+30
answer: -x-10/(x+8)(x+6)
2. x-4/x^2-8x+16 + 3x+21/x^2+12x+35
answer: 4x-7/(x-4)(x+5)
i don't uderstand how to do it. can you please show me the steps. thanks
I am guessing since your typing without parentheses is ambiguous
(x+3)/(x^2+11x+24) - (2x+10)/(x^2+11x+30)
(x+3)/[(x+8)(x+3)] - 2(x+5)/[(x+5)(x+6)]
1/(x+8) - 2/(x+6)
[ (x+6)-2(x+8) ] /[(x+8)(x+6)]
[-x -10 ] / [(x+8)(x+6)]
(x-4)/(x^2-8x+16) + 3(x+7)/(x^2+12x+35)
(x-4)/[(x-4)(x-4)] + 3(x+7)/[(x+7)(x+5)]
proceed just like the other one.
thank you so much i get it now !! :)
Great!! You are welcome :)
Note, it talks about restrictions. Watch out for zeros in denominators.
Certainly! To simplify the given rational expressions and state any restrictions, we need to combine the fractions over a common denominator and then simplify the resulting expression.
Let's start with the first expression:
1. x + 3 / (x^2 + 11x + 24) - (2x + 10) / (x^2 + 11x + 30)
Step 1: Factor the denominators.
The denominator of the first fraction can be factored as (x + 3)(x + 8), and the denominator of the second fraction can be factored as (x + 5)(x + 6).
Step 2: Obtain the common denominator.
The common denominator for these fractions will be (x + 8)(x + 6)(x + 5).
Step 3: Find the numerators with the common denominator.
For the first fraction, multiply the numerator and denominator by (x + 5) to get (x + 3)(x + 5).
For the second fraction, multiply the numerator and denominator by (x + 8) to get -(2x + 10)(x + 8).
Step 4: Combine the fractions.
Now, we have:
[(x + 3)(x + 5) - (2x + 10)(x + 8)] / [(x + 8)(x + 6)(x + 5)]
Step 5: Simplify the expression.
Expanding and simplifying the numerator, we get:
(x^2 + 5x + 3x + 15) - (2x^2 + 16x + 10x + 80)
= x^2 + 8x + 15 - 2x^2 - 26x - 80
= -x^2 - 18x - 65
Therefore, the simplified expression is:
(-x^2 - 18x - 65) / [(x + 8)(x + 6)(x + 5)]
Now let's move on to the second expression:
2. (x - 4) / (x^2 - 8x + 16) + (3x + 21) / (x^2 + 12x + 35)
Step 1: Factor the denominators.
The denominator of the first fraction is a perfect square and can be factored as (x - 4)(x - 4), and the denominator of the second fraction can be factored as (x + 7)(x + 5).
Step 2: Obtain the common denominator.
The common denominator for these fractions will be (x - 4)(x + 7)(x + 5).
Step 3: Find the numerators with the common denominator.
For the first fraction, since the numerator is already (x - 4), it is in the correct form.
For the second fraction, we multiply the numerator and denominator by (x - 4) to get (3x + 21)(x - 4).
Step 4: Combine the fractions.
Now, we have:
[(x - 4) + (3x + 21)(x - 4)] / [(x - 4)(x + 7)(x + 5)]
Step 5: Simplify the expression.
Expanding and simplifying the numerator, we get:
(x - 4 + 3x^2 - 12x + 21x - 84)
= (3x^2 + 10x - 84)
Therefore, the simplified expression is:
(3x^2 + 10x - 84) / [(x - 4)(x + 7)(x + 5)]
Now for the restrictions, if any.
To determine restrictions, we need to look for values of x that make any of the denominators equal to zero.
For the first expression, the denominators are:
(x + 8)(x + 6)(x + 5)
Therefore, the restrictions are x ≠ -8, x ≠ -6, and x ≠ -5.
For the second expression, the denominators are:
(x - 4)(x + 7)(x + 5)
Therefore, the restriction is x ≠ 4.
I hope this helps clarify the steps for simplifying these rational expressions and determining any restrictions. Let me know if you have any further questions!