Question Type 3: Applying polynomial division for rational functions Exercises

Divide $x^4 - 5x^3 + 2x^2 + x - 3$ by $x^2 - x + 1$ and express the result as $Q(x) + \frac{R(x)}{x^2 - x + 1}$.

Divide $2x^3 + 3x^2 - x + 5$ by $x + 2$ and write the result as $Q(x) + \frac{R}{x+2}$.

Divide $x^5 - x^4 + x^3 - x^2 + x - 1$ by $x^2 + 1$ and express the result in the form $Q(x) + \frac{R(x)}{x^2+1}$.

Consider the function $y = \frac{3x^4 - 4x^3 + 5}{x^2 - x - 2}$. Use polynomial division to find the quotient and remainder.

Divide $x^3 - 4x^2 + 7x - 10$ by $x - 3$ and express the answer as $Q(x) + \frac{R}{x-3}$.

Express the rational function $\frac{x^3 + 2x^2 - x + 4}{x - 1}$ in the form $Q(x) + \frac{A}{x - 1}$ by polynomial division.

Perform the division $\frac{4x^3 - x^2 + 2x - 3}{2x - 1}$ and express the result in the form $Q(x) + \frac{R}{2x - 1}$, stating any domain restriction.

Divide $2x^4 + 3x^3 - x^2 + 4x - 5$ by $x^2 + 2x - 3$ and give the quotient and remainder.

Divide $5x^3 + 2x^2 - 7$ by $x + 2$ and find the quotient and remainder.

Use polynomial division to write $\frac{3x^4 -2x^3 + x -6}{x^2 + x -2}$ as $Q(x) + \frac{R(x)}{x^2 + x -2}$.

Compute the division of $3x^3 - x^2 + 4x - 8$ by $x - 2$ and give your answer as $Q(x) + \dfrac{R}{x-2}$.

Perform the polynomial division of $5x^2 + 6x + 9$ by $x - 1$ and express the result in the form $Q(x) + \frac{R}{x-1}$.

Find the equation of the slant asymptote of the function $f(x) = \frac{2x^2 - x + 1}{x + 1}$ by performing polynomial division.

Express $\frac{x^3+x}{x^2+1}$ in the form $Q(x)+\frac{R(x)}{x^2+1}$ by polynomial division.

Divide $x^4 + x^3 - x - 1$ by $x^2 - 1$, simplify fully and give quotient and remainder.