math question ... needing answers tried for hours?

Question

determine the total area of the trapazoid
A=1/2(base1+base2)height

1/2[x-2/1+X+12x-9/2x^2-x-3](3-2x/x3+27)

Express your answer as a rational expression (single fraction) in factored form...

THANKS GUYS!!!

Answer 1

If I understand your question, you have the following:
base 1 = (x-2) / (1+x)
base 2 = (12x - 9) / (2x² - x - 3)
height = (3 - 2x) / (x^3 + 27)

So the area would be:
1 ... (x - 2) .... (12x - 9) ........ (3 - 2x)
-- ( ---------- + -------------- ) x --------------
2 ... (1 + x) .. (2x² - x - 3) .... (x^3 + 27)

First you can factor the denominator in base 2 as:
12x - 9
------------------
(2x - 3)(x + 1)

So to get the same denominator for base 1, you would multiply top and bottom by (2x - 3):
(2x - 3)(x - 2)
------------------
(2x - 3)(x + 1)

Now you can add base 1 and base 2:
(2x - 3)(x - 2) + 12x - 9
---------------------------------
(2x - 3)(x + 1)

Expand out the numerator:
(2x² - 4x - 3x + 6) + 12x - 9
------------------------------------
(2x - 3)(x + 1)

Group like terms in the numerator:
(2x² - 4x - 3x + 12x + 6 - 9)
------------------------------------
(2x - 3)(x + 1)

Simplify:
(2x² + 5x - 3)
------------------
(2x - 3)(x + 1)

Factor the numerator:
(2x - 1)(x + 3)
------------------
(2x - 3)(x + 1)

So far you have:
1 ... (2x - 1)(x + 3) ....... (3 - 2x)
-- ( -------------------- ) x --------------
2 ... (2x - 3)(x + 1) .... (x^3 + 27)

Notice how 3 - 2x = -(2x - 3), so replace:
1 ... (2x - 1)(x + 3) ....... -(2x - 3)
-- ( -------------------- ) x --------------
2 ... (2x - 3)(x + 1) .... (x^3 + 27)

You can cancel the 2x - 3:
1 ... (2x - 1)(x + 3) ....... -1
-- ( -------------------- ) x --------------
2 ......... (x + 1) ........... (x^3 + 27)

Using the sum of cubes rule, factor x^3 + 27 as:
(x + 3)(x² - 3x + 9)

The x + 3 terms cancel and you get:
1 ... (2x - 1) ....... -1
-- x ------------ x ----------------
2 .... (x + 1) ...... (x² - 3x + 9)

Multiply this across to get your final answer:
........ (-2x + 1)
--------------------------
2(x + 1)(x² - 3x + 9)