5.1 Rational operations practice 1

If [math]\displaystyle{\frac{1}{x+1}-\frac{x}{x^2-x+1}}[/math] can be written as [math]\dfrac{ax+b}{x^3+1}[/math] where [math]a[/math] and [math]b[/math] are integers, what is the value of [math]a+b[/math]?

If [math]\displaystyle{\frac{3}{x+2}+\frac{2x+1}{x^2-2x-8}}[/math] can be written as [math]\dfrac{p(x)}{q(x)}[/math] where [math]p(x)[/math] and [math]q(x)[/math] are polynomials, what is the minimum degree of [math]q(x)[/math]?

If [math]\displaystyle{\frac{2+x}{x}\div\frac{x-2}{x^3-x}}=\dfrac{p(x)}{q(x)}[/math] where [math]p(x)[/math] and [math]q(x)[/math] are polynomials, what is [math]\deg(p(x))-\deg(q(x))[/math]?

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

If [math]\dfrac{4x^2+7}{x^3-1}-\dfrac{4}{x-1}=\dfrac{p(x)}{q(x)}[/math] where [math]p(x)[/math] and [math]q(x)[/math] are polynomials, how many terms does [math]p(x)[/math] have?

If [math]f(x)=\dfrac{x^2-4x+3}{x^2+4x+3}\cdot\dfrac{x^2+6x+9}{x^2-6x+9}[/math], how many possible values of [math]n[/math] such that [math]f(n)[/math] is undefined?

If [math]\dfrac{1}{x^2+x}+\dfrac{1}{x^2-x}-\dfrac{1}{x^2-1}[/math] can be written as [math]\dfrac{p(x)}{q(x)}[/math] where [math]p(x)[/math] and [math]q(x)[/math] are polynomials, what is the minimum degree of [math]p(x)[/math]?

If the expression [math]\dfrac{x^3+x^2+4x+6}{x-1}[/math] can be written as [math]x^2+2x+6+\dfrac{a}{x-1}[/math] where [math]a[/math] is a constant, what is the value of [math]a[/math]?

Consider two functions [math]f(x)=\dfrac{4x^2+4x}{x+1}[/math] and [math]g(x)=4x[/math]. Which of the following statements is/are true?

(Calculator) If [math]\displaystyle{\frac{2x^2-7x+14}{x^2+8}}=a-\dfrac{bx+c}{x^2+8}[/math] where [math]a, b[/math] and [math]c[/math] are integers, what are [math]a, b[/math], and [math]c[/math]?