I'm not sure how to divide rational expressions:
((1/x+1)+(1/x+2))/((1/x+2)+(1/(x+3))
-I got ((2x+3)(x^2+5x+6)/(2x+5)(x^2+3x+2)
but it was wrong? Please help!
Who said it was wrong ?
You are right up to this point, but your answer can be reduced.
((2x+3)(x^2+5x+6)/(2x+5)(x^2+3x+2)
= [(2x+3)(x+2)(x+3)/[(2x+5)(x+2)(x+1)]
= [(2x+3)(x+3)/[(2x+5)(x+1)] after canceling the x+2
you should also include the restriction that x not equal to -2
To divide rational expressions, follow these steps:
Step 1: Simplify both the numerator and the denominator by factoring and canceling out common factors.
Let's simplify the expression you provided:
Numerator:
Start by simplifying the expression (1/x + 1) by finding a common denominator.
The common denominator is (x + 1)(x + 2), so we can rewrite 1/x + 1 as (x + 2)/(x(x + 2)) + (x(x + 1))/(x(x + 1)(x + 2)).
Combine the fractions with the same denominator:
[(x + 2) + x(x + 1)] / (x(x + 1)(x + 2)).
Simplify by expanding x(x + 1):
[(x + 2) + (x^2 + x)] / (x(x + 1)(x + 2)).
Simplify further by combining like terms:
(2x^2 + 3x + 2) / (x(x + 1)(x + 2)).
Denominator:
Again, find a common denominator for the expression (1/x + 2) + (1/(x + 3)).
The common denominator is x(x + 3), so we can rewrite 1/x + 2 as 2(x + 3)/(x(x + 3)) and 1/(x + 3) as x/(x(x + 3)).
Combine the fractions with the same denominator:
[2(x + 3) + x] / (x(x + 3)).
Simplify by removing the parentheses:
[2x + 6 + x] / (x(x + 3)).
Combine like terms:
(3x + 6) / (x(x + 3)).
Step 2: Rewrite the division as multiplication by taking the reciprocal of the second rational expression.
Reciprocal of (3x + 6) / (x(x + 3)) is (x(x + 3)) / (3x + 6).
Now, we have:
(2x^2 + 3x + 2) / (x(x + 1)(x + 2)) * (x(x + 3)) / (3x + 6).
Step 3: Cancel out common factors and simplify.
To cancel out the common factors, look for factors that appear both in the numerator and the denominator. In this case, we can cancel out (x + 2) from the numerator and denominator and (3x + 6) from the numerator and denominator.
Simplifying the expression, we have:
[(2x^2 + 3x + 2) * x(x + 3)] / [(x(x + 1)) * (3x + 6)].
Further simplification can be done by factoring the quadratic expression in the numerator:
[(2x + 1)(x + 2) * x(x + 3)] / [(x(x + 1)) * (3(x + 2))].
Now we can cancel some more common factors:
[(2x + 1) * x(x + 3)] / [x * (3(x + 2))].
Finally, we get:
(2x + 1)(x + 3) / (3(x + 2)).
So, the correct simplified form of the expression is:
(2x + 1)(x + 3) / (3(x + 2)).