Polynomial Equations (AI SL)
A polynomial equation sets an expression in powers of \(x\) equal to zero, and asks for the values of \(x\) that make it true - its roots. IB AI SL exams never ask you to factorise or solve these by hand: the syllabus expects the GDC's equation solver to do the algebra while you focus on setting up and interpreting the equation. This page covers that process with worked examples and the mistakes that trip students up. It's part of the broader Systems & Polynomial Equations topic.
29 questions on this sub-topic.
Solving with technology
Covered under IB syllabus reference SL1.8. There's no algebraic formula to memorise here - the skill is entering the equation correctly and reading off every root your GDC returns. See the GDC guide for calculator-specific steps.
General polynomial equation
\(f(x)=0\)
Not in the formula booklet - solved entirely by GDC, whatever the degree of \(f(x)\).
Number of roots
Degree \(n \Rightarrow\) up to \(n\) real roots
A quadratic has up to 2 roots, a cubic up to 3. Check your GDC's answer list matches the degree before you stop.
Need the full syllabus wording and formula-booklet reference table? See Systems & Polynomial Equations.
Worked examples
Solve \(2x^2-5x-3=0\) using technology.
(a)(i) Give the value with \(x<1\).
(a)(ii) Give the value with \(x>1.\)
Worked solution
(a)(i) Enter the coefficients into the polynomial solver. M1 \(x=-0.5\) A1
(a)(ii) or \(x=3.\) A1
An open box has volume \(V=x(20-2x)(15-2x)\) cm\(^3\).
(a)(i) Find the value of \(x<3\) for which the volume is 300 cm\(^3\).
(a)(ii) Find the value of \(x>3\).
Worked solution
Set \(x(20-2x)(15-2x)=300.\) M1
Solving graphically/with technology gives M1
\(x\approx1.45\) cm or \(x\approx4.47\) cm (both valid for \(0<x<7.5\)). A1 A1
Solve \(x^4-5x^2+4=0\) using technology.
(a)(i) Give the value with \(x<-1.5\).
(a)(ii) Give the value with \(-1.5<x<0\).
(a)(iii) Give the value with \(0<x<1.5\).
(a)(iv) Give the value with \(x>1.5.\)
Worked solution
(a)(i) Enter the polynomial. M1 \(x=-2\) A1
(a)(ii) \(x=-1\) A1
(a)(iii) \(x=1\) A1
(a)(iv) \(x=2.\) A1
Common mistakes
- Stopping after the first root. A cubic can have up to three real roots - if your GDC's solver only shows one, check whether it has a "next solution" option or re-graph to find the others before you submit an answer.
- Losing a solution to a domain restriction. Problems like the box-volume example only make sense for \(x\) in a certain range (here \(0
- Mistyping coefficients, including the sign. A single dropped minus sign in the solver changes every root it returns - re-read the equation from the question before pressing solve, not just the numbers you remember.
Ready to practise properly?
48 polynomial-equation questions, marked instantly like the real exam.
Quick answers
How do I solve a polynomial equation for IB AI SL?
Enter the coefficients into your GDC's polynomial root finder, or plot the equation and read off where it crosses the \(x\)-axis. No specific method is required in the exam.
How many solutions can a polynomial equation have?
Up to as many real roots as its degree - a quadratic has up to 2, a cubic up to 3. Always check your GDC's list of roots against the degree of the polynomial.