sums and differences of rational algebraic expression answer the following 2x/x^2-2x-15 - x-2/3x^2+9 another is t/2rs^2 - 2r/3rst + 3s/30r^2t another is 3x/2x-3 - 2x/2x+3 + 3/4x^2-9 please answer this i need it tomorrow morning 3:00 please answer it i will WAIT FOR IT!
To simplify rational algebraic expressions, follow these steps:
1. Factor the denominators of each expression.
- For the first expression:
x^2 - 2x - 15 = (x - 5)(x + 3)
3x^2 + 9 = 3(x^2 + 3)
- For the second expression:
2rs^2, 3rst, and 30r^2t are already factored.
- For the third expression:
2x - 3 = (2x - 3)
2x + 3 = (2x + 3)
4x^2 - 9 = (2x - 3)(2x + 3)
2. Find the least common denominator (LCD) by taking the product of all the unique factors from the denominators.
- For the first expression, the LCD is: 3(x - 5)(x + 3)
- For the second expression, the LCD is: 2rs^2 * 3rst * 30r^2t = 180r^2s^2t^2
- For the third expression, the LCD is: (2x - 3)(2x + 3)(4x^2 - 9)
3. Rewrite each expression with the LCD as the common denominator.
- For the first expression:
(2x / (x^2 - 2x - 15)) - ((x - 2) / (3x^2 + 9)) =
(2x * 3) / (3(x - 5)(x + 3)) - (x - 2) * (x - 5) / (3(x^2 + 3)) =
(6x) / (3(x - 5)(x + 3)) - (x - 2)(x - 5) / (3(x^2 + 3))
- For the second expression:
(t / (2rs^2)) - (2r / (3rst)) + (3s / (30r^2t)) =
(t * 3r * 10t * s) / (180r^2s^2t^2) - (2r * 20r * s * t) / (180r^2s^2t^2) + (3s * rs * 6) / (180r^2s^2t^2) =
(30t^2s - 40rst + 18rs) / (180r^2s^2t^2)
- For the third expression:
(3x / (2x - 3)) - (2x / (2x + 3)) + (3 / (4x^2 - 9)) =
(3x * (2x + 3)(4x^2 - 9) / ((2x - 3)(2x + 3)(4x^2 - 9))) - (2x * (2x - 3)(4x^2 - 9) / ((2x - 3)(2x + 3)(4x^2 - 9))) + (3 * (2x - 3)(2x + 3) / ((2x - 3)(2x + 3)(4x^2 - 9))) =
(6x^3 + 9x^2 - 18x - 27) / ((2x - 3)(2x + 3)(4x^2 - 9))
These are the simplified forms of the given rational algebraic expressions. Please note the steps provided above for simplifying the expressions.