Question

100 POINTS!!

6. Add (x + 2/x2 - 1)) + (3/x + 1) + (4/x + 1)
7. Simplify ((c3 + 8)/(c2 - 4)) * ((c2 - 4c + 4)/(c2 - 2c + 4))
8. Subtract ((4a2 - 1)/a2 - 4)) - ((2a - 1)/(a - 2)
9. p(x) = (1/x + 1) and q(x) = (1/x - 1) perform the operation and show that it results in another rational expression p(x) * q(x)
10. Simplify ((r2 - 4s2)/(r + 2s)) / (r + 2s)
11. p(x) = (1/x + 1) and q(x) = (1/x - 1) perform the operation and show that it results in another rational expression p(x) + q(x)
12. Simplify ((y2 - 16)/(2y + 6)) * ((y + 3)/(y - 4))

Answer 1

The solutions of the given expressions would be as follows:

1. 8x - 5 / (x^2 - 1)

2. (c^2 - 2c + 4) / (c - 2)

3. (2a^2 - 3a + 1) / ((a^2 - 4)(a - 2))

4. 1 / (x^2 - 1)

5. (r^2 - 4s^2) / (r + 2s)^2

6. 2x / (x^2 - 1)

7. (y + 4) / 2

6. Combine like terms:

x + 7/x + 2/x^2 - 1 + 1 + 1

x + 7/x + 2/x^2 + 1

Combine x and constant terms:

x + 7/x + 2/x^2 + 1

Final simplified expression:

8x - 5 / x^2 - 1

7. Factor and simplify:

(c + 2)(c^2 - 2c + 4) / (c + 2)(c - 2)

Cancel common factors:

c^2 - 2c + 4 / c - 2

8. Combine into a single fraction:

(4a^2 - 1) - (2a - 1)(a + 2) / (a^2 - 4)(a - 2)

Simplify the numerator:

(2a^2 - 3a + 1) / (a^2 - 4)(a - 2)

9. Multiply p(x) and q(x):

1 / (x + 1) * 1 / (x - 1)

p(x) * q(x) = 1 / (x^2 - 1)

10. Divide by multiplying by the reciprocal:

(r^2 - 4s^2) / (r + 2s) / (r + 2s)

Cancel common factors:

(r^2 - 4s^2) / (r + 2s)^2

11. Add p(x) and q(x):

1 / (x + 1) + 1 / (x - 1)

Combine into a single fraction:

(x - 1 + x + 1) / (x + 1)(x - 1)

Simplify:

2x / (x^2 - 1)

12. Factorize and simplify:

(y - 4)(y + 4) / 2(y + 3)

Cancel out common factors:

(y + 4) / 2