Add or subtract the following rational expressions. Be sure to look for factors before trying to determine a common denominator, and simplify your answers, if possible.
16x-12/4x^2+5x-6-3/x+2
From the many ways your ambiguous way of typing could be interpreted , I will choose
(16x-12)/(4x^2+5x-6) - 3/(x+2)
= 4(x-3)/( (x+2)(4x-3) ) - 3/(x+2)
= (4(x-3) - 3(4x-3))/((x+2)(4x-3))
= (4x - 12 - 12x + 9)/((x+2)(4x-3))
= (-8x - 3)/((x+2)(4x-3))
To add or subtract rational expressions, you need to find a common denominator. Let's simplify and factor the expressions first:
The expression 16x-12 can be factored out 4: 4(4x - 3).
The expression 4x^2+5x-6 can be factored as (4x - 3)(x + 2).
Now, let's rewrite the expression:
(4(4x - 3))/(4x^2+5x-6) - 3/(x + 2)
The common denominator is (4x^2+5x-6)(x + 2).
Next, let's rewrite the expression with the common denominator:
[(4(4x - 3))/(4x^2+5x-6)] * [(x + 2)/(x + 2)] - [3/(x + 2)] * [(4x^2+5x-6)/(4x^2+5x-6)]
Simplifying this expression gives us:
(4(4x - 3)(x + 2) - 3(4x^2+5x-6))/(4x^2+5x-6)(x + 2)
Now let's expand and simplify this expression:
(16x^2 + 8x - 12x - 24 - 12x^2 - 15x + 18)/(4x^2+5x-6)(x + 2)
Combining like terms in the numerator:
(4x^2 + x - 6)/(4x^2+5x-6)(x + 2)
After factoring the numerator and the denominator, it is evident that there are no common factors to cancel out. Therefore, this expression is simplified and cannot be simplified any further.
The final result is: (4x^2 + x - 6)/(4x^2+5x-6)(x + 2)
To add or subtract rational expressions, you need to find a common denominator and then combine the terms.
Let's simplify the expressions first.
The first rational expression can be simplified as follows:
16x - 12 / 4x^2 + 5x - 6
The numerator doesn't have any common factors, so we'll focus on factoring the denominator:
4x^2 + 5x - 6
To factor this quadratic expression, we need to find two numbers whose product is equal to the product of the coefficient of x^2 (4) and the constant term (-6), and whose sum is equal to the coefficient of x (5).
For this expression, the numbers that meet these conditions are 4 and -3.
4x^2 + 5x - 6 = (4x - 3)(x + 2)
So, the first rational expression becomes:
(16x - 12) / (4x - 3)(x + 2)
Now, let's simplify the second rational expression:
-3 / x + 2
Now that we have simplified both expressions, we can find a common denominator and combine the terms.
To find the common denominator, multiply the denominators of both expressions together:
(4x - 3)(x + 2)
Now, let's rewrite the expressions with the common denominator:
[(16x - 12)(x + 2)] / [(4x - 3)(x + 2)] - (-3)(4x - 3) / [(4x - 3)(x + 2)]
To combine the terms, we can distribute and simplify:
[(16x^2 + 32x - 24) - (12x - 9)] / [(4x - 3)(x + 2)]
Simplifying further:
(16x^2 + 32x - 24 - 12x + 9) / (4x^2 + 8x - 3x - 6)
Combining like terms:
(16x^2 + 20x - 15) / (4x^2 + 5x - 6)
At this point, we can check if the expression can be simplified further. In this case, it cannot be simplified any further.
So, the sum/difference of the given rational expressions is:
(16x^2 + 20x - 15) / (4x^2 + 5x - 6)