5.1 Rational operations (in class)

Which of the following expressions is equivalent to [math]1+\displaystyle{\frac{x^2-1}{x^2+1}}[/math]?

If [math]\displaystyle{\frac{6-2x}{4x}\cdot\frac{(x-1)^{-1}}{x^2-6x-9}}=\dfrac{p(x)}{q(x)}[/math] where [math]p(x)[/math] and [math]q(x)[/math] are polynomials, which of the following expressions is not a factor of [math]q(x)[/math]?

If [math]\left(\dfrac{x}{x+2}+\dfrac{2}{x-2}\right)\div\left(\dfrac{x}{x+2}-\dfrac{2}{x-2}\right)[/math] can be written as [math]\dfrac{p(x)}{q(x)}[/math] where [math]p(x)[/math] and [math]q(x)[/math] are quadratic functions, which of the following could be [math]q(x)[/math]?

If the expression [math]\dfrac{x^3+4x^2-19x+27}{x^2+6x-7}[/math] can be written as [math]ax+b+\dfrac{13}{x^2+6x-7}[/math] where [math]a[/math] and [math]b[/math] are constants, what are [math]a[/math] and [math]b[/math]?

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{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]?

Find [math]a, b, c[/math], and [math]d[/math] such that [math]\dfrac{\frac{3x}{x-2}-1}{\frac{6}{x-2}-\frac{5}{x+3}}=\dfrac{ax^2+bx+c}{x+d}[/math] is an identity.

If [math]\dfrac{3x^2-4x+1}{\frac{3x^2+2x-1}{3x+1}}=\dfrac{p(x)}{x+1}[/math] where [math]p(x)[/math] is a polynomial, what is [math]p(x)[/math]?