IB Maths AA HL · Unit 2: Functions
IB Maths AA HL Polynomials and Rational Functions Questions
Exam-style IB Maths AA HL polynomials and rational functions questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 33 questions
- Paper 1: 29
- Paper 2: 4
- 7 easy
- 12 medium
- 9 hard
- 5 starter
- 3 worked examples
Practise Polynomials and Rational Functions questions →
AA HL formula booklet
What's examined in AA HL polynomials and rational functions
The question bank covers these polynomials and rational functions question types (number of questions in brackets):
- Polynomial Roots & Factors (22)
- Rational Function Asymptotes (9)
- Rational Function Inequalities (2)
Polynomials and Rational Functions worked examples
Worked example 1: The Remainder Theorem · easy
Let $P(x) = 2x^3 - x^2 + kx - 4$. Given that dividing $P(x)$ by $(x - 2)$ leaves a remainder of $10$, find the exact value of the constant $k$.
1. State the Remainder Theorem: If a polynomial $P(x)$ is divided by $(x - a)$, the remainder is $P(a)$. Here, $a = 2$.
2. Set up the equation using the given remainder: $P(2) = 10$.
3. Substitute $x = 2$ into the polynomial: $2(2)^3 - (2)^2 + k(2) - 4 = 10$.
4. Evaluate the powers first: $2(8) - 4 + 2k - 4 = 10$.
5. Simplify the constants: $16 - 8 + 2k = 10 \implies 8 + 2k = 10$.
6. Solve for $k$: $2k = 2 \implies \mathbf{k = 1}$.
Examiner tip: Ensure the substitution correctly applies exponents before multiplication (e.g., $2(2)^3 = 2 \times 8 = 16$). A common error is multiplying the coefficient and base first (getting $4^3 = 64$), which breaks the order of operations.
Worked example 2: Finding oblique asymptotes · medium
Find the exact equation of the oblique (slant) asymptote for the rational function $f(x) = \frac{x^2 + 5x - 2}{x + 2}$.
1. Recognize that an oblique asymptote exists because the degree of the numerator (2) is exactly one higher than the degree of the denominator (1).
2. Perform algebraic long division or synthetic division. Divide $x^2 + 5x - 2$ by $x + 2$.
3. Calculate the first term: $x^2 \div x = x$. Multiply $x(x+2) = x^2+2x$ and subtract to leave $3x - 2$.
4. Calculate the second term: $3x \div x = 3$. Multiply $3(x+2) = 3x+6$ and subtract to leave a remainder of $-8$.
5. Rewrite the function: $f(x) = x + 3 - \frac{8}{x+2}$.
6. Conclude: As $x \to \pm\infty$, the fractional remainder $\frac{8}{x+2} \to 0$. The oblique asymptote is the linear quotient, $\mathbf{y = x + 3}$.
Examiner tip: Synthetic division is a fast alternative here, but students must remember that the oblique asymptote is given by the quotient alone; the remainder must be entirely discarded when stating the asymptote's equation.
Worked example 3: Sum and product of roots · hard
The cubic equation $2x^3 - 4x^2 + 5x - 7 = 0$ has complex roots $\alpha, \beta, \gamma$. Find the exact value of $\alpha^2 + \beta^2 + \gamma^2$.
1. Identify the standard properties of roots for a cubic $ax^3+bx^2+cx+d=0$. The sum of the roots is $\sum \alpha = \alpha + \beta + \gamma = -\frac{b}{a}$.
2. Calculate the sum: $\alpha + \beta + \gamma = -\frac{-4}{2} = 2$.
3. Identify the sum of the product of pairs: $\sum \alpha\beta = \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} = \frac{5}{2}$.
4. Recall the algebraic expansion identity for a trinomial squared: $(\alpha + \beta + \gamma)^2 = \alpha^2 + \beta^2 + \gamma^2 + 2(\alpha\beta + \beta\gamma + \gamma\alpha)$.
5. Substitute the known values into the identity: $2^2 = \alpha^2 + \beta^2 + \gamma^2 + 2\left(\frac{5}{2}\right)$.
6. Simplify and solve for the sum of squares: $4 = \alpha^2 + \beta^2 + \gamma^2 + 5 \implies \mathbf{\alpha^2 + \beta^2 + \gamma^2 = -1}$.
Examiner tip: If you get a negative sum of squares, don't panic! A negative result proves that at least two of the roots must be complex numbers (specifically a complex conjugate pair), which is a great self-check for HL students.
Try these IB Maths AA HL polynomials and rational functions questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 3 marks · Paper 1
Let the polynomial be \(P(x) = 2x^3 - x^2 + kx - 4\). When \(P(x)\) is divided by \((x - 2)\), the remainder is \(10\). Find the value of the constant \(k\).
Question 2 · medium · 6 marks · Paper 1
A polynomial $P(x) = x^3 + ax^2 + bx + 10$ has a factor $(x-2)$. When $P(x)$ is divided by $(x+1)$, the remainder is $12$. Find the values of $a$ and $b$.
Attempt it and see the mark scheme →
Question 3 · hard · 7 marks · Paper 2
Consider the rational function \(y = \frac{x^2 + 2x + 5}{x + 1}\) for \(x \neq -1\).
By rearranging the equation into a quadratic in \(x\) and analyzing its discriminant, find the exact range of the function.
All 33 polynomials and rational functions questions with mark schemes →
FAQ
How many IB Maths AA HL polynomials and rational functions questions are there?
There are 33 exam-style polynomials and rational functions questions in the AA HL question bank (Paper 1: 29 · Paper 2: 4), graded 7 easy, 12 medium, 9 hard, 5 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is polynomials and rational functions on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 29 · Paper 2: 4. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA HL Unit 2 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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