((5)/((6x^2)+46x-16))+((2)/((6x^2)+57x+72)

Question

adding and subtracting rational expressions

 

Answer 1

     (5/(6x² + 46x - 16)) + (2/(6x² + 57x + 72))

Recall that in order to add fractions, the fractions need to share a common denominator. In this case, the least common denominator that these two fractions will share is a polynomial. To find this LCD, we first need to factor out the polynomials in the denominator of each rational expression:

     6x² + 46x - 16 = (x + 8)(6x - 2)

     6x² + 57x + 72 = (x + 8)(6x + 9)

For these two polynomials to be equal, we have to multiply the first one by the factor that is different in the 2nd polynomial (i.e., multiply by '6x + 9') and multiply the 2nd polynomial by the factor that is different in the 1st polynomial (i.e., multiply by '6x - 2'). To get this common denominator without changing the value of each rational expression, we multiply the each numerator by the same factor we multiplied its denominator with. That is,

     (5/((6x - 2)(x + 8))) + (2/((6x + 9)(x + 8)))

     ((5(6x + 9))/((x + 8)(6x - 2)(6x + 9))) + ((2(6x - 2)/((x + 8)(6x + 9)(6x - 2)))

     ((30x + 45)/((x + 8)(6x - 2)(6x + 9))) + ((12x - 4)/((x + 8)(6x + 9)(6x - 2)))

Now that the 2 rational expression have a common denominator, we can combine their numerators and keep the common denominator:

     ((30x + 45)+(12x - 4))/((x + 8)(6x - 2)(6x + 9))

     (30x + 12x + 45 - 4)/((x + 8)(6x - 2)(6x + 9))

     (42x + 41)/((x + 8)(6x - 2)(6x + 9))

Since neither the numerator nor the denominator can be factored out any further, and thus nothing will cancel out, this is the solution in its most reduced form. You can multiply out the factors in the denominator to get a polynomial or leave it in its factored form, but the solution remains the same.