\( \begin{array}{llll}\text { 49. } T(x)=\frac{x^{3}}{x^{4}-1} & \text { 50. } P(x)=\frac{4 x^{2}}{x^{3}-1} & \text { 51. } Q(x)=\frac{2 x^{2}-5 x-12}{3 x^{2}-11 x-4} & \text { 52. } F(x)=\frac{x^{2}+6 x+5}{2 x^{2}+7 x+5} \\ \text { 53. } R(x)=\frac{6 x^{2}+7 x-5}{3 x+5} & \text { 54. } R(x)=\frac{8 x^{2}+26 x-7}{4 x-1} & \text { 55. } G(x)=\frac{x^{4}-1}{2} & \text { 56. } F(x)=\frac{x^{4}-16}{x^{2}-2 x}\end{array} \)

Sure! Let's simplify and factor each of the given rational functions: --- ### **49.** \( T(x) = \frac{x^{3}}{x^{4} - 1} \) **Factorization:** - Denominator: \( x^4 - 1 = (x^2)^2 - 1^2 = (x^2 - 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1) \) **Simplified Form:** \[ T(x) = \frac{x^{3}}{(x - 1)(x + 1)(x^2 + 1)} \] --- ### **50.** \( P(x) = \frac{4x^{2}}{x^{3} - 1} \) **Factorization:** - Denominator: \( x^3 - 1 = (x - 1)(x^2 + x + 1) \) **Simplified Form:** \[ P(x) = \frac{4x^{2}}{(x - 1)(x^2 + x + 1)} \] --- ### **51.** \( Q(x) = \frac{2x^{2} - 5x - 12}{3x^{2} - 11x - 4} \) **Factorization:** - Numerator: \( 2x^2 - 5x - 12 = (2x + 3)(x - 4) \) - Denominator: \( 3x^2 - 11x - 4 = (3x + 1)(x - 4) \) **Simplified Form:** \[ Q(x) = \frac{(2x + 3)(x - 4)}{(3x + 1)(x - 4)} = \frac{2x + 3}{3x + 1} \quad \text{(for } x \neq 4\text{)} \] --- ### **52.** \( F(x) = \frac{x^{2} + 6x + 5}{2x^{2} + 7x + 5} \) **Factorization:** - Numerator: \( x^2 + 6x + 5 = (x + 1)(x + 5) \) - Denominator: \( 2x^2 + 7x + 5 = (x + 1)(2x + 5) \) **Simplified Form:** \[ F(x) = \frac{(x + 1)(x + 5)}{(x + 1)(2x + 5)} = \frac{x + 5}{2x + 5} \quad \text{(for } x \neq -1\text{)} \] --- ### **53.** \( R(x) = \frac{6x^{2} + 7x - 5}{3x + 5} \) **Factorization:** - Numerator: \( 6x^2 + 7x - 5 = (2x - 1)(3x + 5) \) **Simplified Form:** \[ R(x) = \frac{(2x - 1)(3x + 5)}{3x + 5} = 2x - 1 \quad \text{(for } x \neq -\frac{5}{3}\text{)} \] --- ### **54.** \( R(x) = \frac{8x^{2} + 26x - 7}{4x - 1} \) **Factorization:** - Numerator: \( 8x^2 + 26x - 7 = (4x - 1)(2x + 7) \) **Simplified Form:** \[ R(x) = \frac{(4x - 1)(2x + 7)}{4x - 1} = 2x + 7 \quad \text{(for } x \neq \frac{1}{4}\text{)} \] --- ### **55.** \( G(x) = \frac{x^{4} - 1}{2} \) **Factorization:** - Numerator: \( x^4 - 1 = (x^2)^2 - 1^2 = (x^2 - 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1) \) **Simplified Form:** \[ G(x) = \frac{(x - 1)(x + 1)(x^2 + 1)}{2} \] --- ### **56.** \( F(x) = \frac{x^{4} - 16}{x^{2} - 2x} \) **Factorization:** - Numerator: \( x^4 - 16 = (x^2)^2 - 4^2 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4) \) - Denominator: \( x^2 - 2x = x(x - 2) \) **Simplified Form:** \[ F(x) = \frac{(x - 2)(x + 2)(x^2 + 4)}{x(x - 2)} = \frac{(x + 2)(x^2 + 4)}{x} \quad \text{(for } x \neq 0 \text{ and } x \neq 2\text{)} \] --- If you have any specific questions or need further assistance with these functions, feel free to ask!