How do you simplify #(r^2+25) / (8r^3 -27) - ( 3r+2) /(2r-3)#?

Question

Eva Adamson · Accepted Answer

#(-12r^3-25r^2-39r+7)/[(2r-3)(4r^2+6r+9)#

First, factor #8r^3-27# into #(2r-3)(4r^2+6r+9)# using #(a^3-b^3)=(a-b)(a^2+ab+b^2)#.

We now get

#(r^2+25)/[(2r-3)(4r^2+6r+9)]-(3r+2)/(2r-3)#

Multiply by #(4r^2+6r+9)# in the second expression to get

#[(3r+2)(4r^2+6r+9)]/[(2r-3)(4r^2+6r+9)#

Now you can combine to get

#{(r^2+25)-(3r+2)(4r^2+6r+9)}/[(2r-3)(4r^2+6r+9)#

Expanding gives you

#{(r^2+25)-(12r^3+26r^2+39r+18)}/[(2r-3)(4r^2+6r+9)#

Last step is to simplify and you will get

#(-12r^3-25r^2-39r+7)/[(2r-3)(4r^2+6r+9)#

Jameson Allen · Answer

undefined To simplify the expression (r^2+25) / (8r^3 -27) - (3r+2) /(2r-3), we need to find a common denominator and combine the fractions. The common denominator is (8r^3 - 27)(2r - 3).

To combine the fractions, we multiply the numerator and denominator of the first fraction by (2r - 3), and the numerator and denominator of the second fraction by (8r^3 - 27).

After simplifying, the expression becomes:
[(r^2 + 25)(2r - 3) - (3r + 2)(8r^3 - 27)] / [(8r^3 - 27)(2r - 3)]

Expanding and simplifying the numerator, we get:
(2r^3 - 3r^2 + 50r - 75 - 24r^4 + 81r + 16r^3 - 54) / [(8r^3 - 27)(2r - 3)]

Combining like terms in the numerator, we have:
(-24r^4 + 18r^3 - 3r^2 + 50r + 81r - 75 - 54) / [(8r^3 - 27)(2r - 3)]

Simplifying further, we get:
(-24r^4 + 18r^3 - 3r^2 + 131r - 129) / [(8r^3 - 27)(2r - 3)]

Therefore, the simplified expression is (-24r^4 + 18r^3 - 3r^2 + 131r - 129) / [(8r^3 - 27)(2r - 3)].