5.1 Rational operations (partial fractions)

Let [math]x[/math] be a positive integer.

(a) Find constants [math]a[/math] and [math]b[/math] such that [math]\dfrac{2}{x^2+2x}=\dfrac{a}{x}+\dfrac{b}{x+2}[/math].

(b) If [math]\dfrac{2}{x(x+2)}+\dfrac{2}{(x+2)(x+4)}+\dfrac{2}{(x+4)(x+6)}+\cdots+\dfrac{2}{(x+98)(x+100)}=\dfrac{1}{200}[/math], what is the value of [math]x[/math]?

(Calculator) If [math]\displaystyle{\frac{x+12}{x^2-x-6}}=\dfrac{a}{x-3}+\dfrac{b}{x+2}[/math] where [math]a[/math] and [math]b[/math] are integers, what is [math]a+b[/math]?

(Calculator) If [math]\displaystyle{\frac{2x+3}{(x+1)^2}}=\dfrac{a}{x+1}+\dfrac{b}{(x+1)^2}[/math] where [math]a[/math] and [math]b[/math] are integers, what is [math]a+b[/math]?

(Calculator) If [math]\displaystyle{\frac{4x^2+9x-4}{(x-1)(x+2)^2}}=\dfrac{a}{x-1}+\dfrac{b}{x+2}+\dfrac{c}{(x+2)^2}[/math] where [math]a, b[/math] and [math]c[/math] are integers, what are [math]a, b[/math], and [math]c[/math]?

(Calculator) If [math]\displaystyle{\frac{5x^2+2x+2}{x^3-1}}=\dfrac{a}{x-1}+\dfrac{bx+c}{x^2+x+1}[/math] where [math]a, b[/math], and [math]c[/math] are integers, what is the value of [math]b[/math]?

(Calculator) If [math]\displaystyle{\frac{-4x+7}{(x+5)^2}}=\dfrac{a}{x+5}+\dfrac{b}{(x+5)^2}[/math] where [math]a[/math] and [math]b[/math] are integers, what is the value of [math]a+b[/math]?

(Calculator) If [math]\displaystyle{\frac{2x^2+3x}{x^2-3x+2}}=a+\dfrac{b}{x-2}+\dfrac{c}{x-1}[/math] where [math]a, b[/math], and [math]c[/math] are integers, what is the value of [math]c[/math]?

If [math]\dfrac{17x-53}{x^2-2x-15}=\dfrac{a}{x+3}+\dfrac{b}{x-5}[/math] where [math]a[/math] and [math]b[/math] are integers, what is the value of [math]b[/math]?

(Calculator) Find the values of [math]A, B, C, D[/math] and [math]E[/math] that satisfy the following equation.$$\dfrac{-4x^4+10x^3+14x^2+6x}{x(x-3)(x-1)(x+1)(x+2)}=\dfrac{A}{x}+\dfrac{B}{x-3}+\dfrac{C}{x-1}+\dfrac{D}{x+1}+\dfrac{E}{x+2}$$