IB Mathematics Analysis and Approaches 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 $Q(x)+\frac{R}{x-2}$.

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.