Solving systems and polynomial equations with technology

IB Mathematics Applications and Interpretation SL· Unit 1 Number and Algebra: Technology-enabled algebraic solving· 12 min read

1. GDC Pre-Configuration for IB AI SL Algebra Tasks★★☆☆☆⏱ 10 min

✓ Calculator OK

Before starting any exam problem, confirm your GDC is set to display 3 significant figures by default, and that the graph window default range is set to x: -10 to 10, y: -10 to 10. This avoids missing negative roots or producing answers outside IB tolerance ranges.

📘 Definition

GDC Polynomial Solver

N/A

A built-in tool that accepts polynomial coefficients as input and returns all real roots without requiring manual factorization

Example:

Inputting coefficients for $2 x^{2} + 5 x - 3 = 0$ returns roots $x = 0.5$ and $x = - 3$

📐 Worked Example

Configure your TI-84 GDC to show 3 s.f. outputs and confirm the default graph window range

1
Press MODE, scroll down to Float, select '3' to set 3 significant figure display
2
Press WINDOW, set Xmin=-10, Xmax=10, Ymin=-10, Ymax=10
3
Press GRAPH to confirm no error messages appear, then save the settings to a custom memory slot

✓ Quick check

Confirm you can complete this setup on your own GDC:

What is the standard IB required number of significant figures for final answers?
- 2 s.f.
- 3 s.f.
- 4 s.f.
- 1 s.f.
Reveal answer
3 s.f. —
All final answers in IB AI SL default to 3 significant figures unless the question explicitly states otherwise.

2. Solving 2x2 and 3x3 Linear Systems with GDC★★★☆☆⏱ 12 min

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For systems with 2 or 3 variables, you do not need to use substitution or elimination methods on Paper 2. The GDC simultaneous equation solver will return exact solutions in seconds, eliminating almost all arithmetic errors.

\begin{cases} 2x + y - z = 1 \\ x - y + 2z = 3 \\ 3x + 2y + z = 4 \end{cases}

📐 Worked Example

Solve the 3x3 linear system shown above using your GDC

1
Open the 'Simultaneous Equation' tool on your GDC, select 3 variables
2
Input the coefficients for each equation exactly as written, including the = value on the right hand side
3
Run the solver to get outputs $x = 1$ , $y = 0$ , $z = 1$
4
Verify by substituting back into all three equations: $2 (1) + 0 - 1 = 1$ , $1 - 0 + 2 (1) = 3$ , and $3 (1) + 0 + 1 = 4$ — all hold

3. Finding Real Roots of Polynomial Equations Up to Degree 4★★★☆☆⏱ 10 min

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For polynomials of degree 3 or higher, manual factorization is almost never expected in IB AI SL. Use the GDC polynomial solver or graph root finder tool to capture all real roots directly.

📐 Worked Example

Find all real roots of the cubic polynomial $f (x) = x^{3} - 4 x^{2} + x + 6$ using your GDC

1
Open the polynomial solver tool, set degree to 3
2
Input coefficients a=1, b=-4, c=1, d=6
3
Run the solver to get outputs $x = - 1$ , $x = 2$ , $x = 3$
4
Cross-check by graphing the function to confirm all 3 x-intercepts appear in your window range

4. Presenting GDC Working for Full Exam Marks★★★★☆⏱ 8 min

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Mark Type	Requirement for GDC Solutions
Accuracy Mark	Final answer rounded to 3 s.f. within IB tolerance range
Method Mark	1 line note that GDC was used, plus 1 intermediate output value
Communication Mark	Explicitly state which variable corresponds to which output value

✓ Quick check

Test your understanding of mark rules:

What is the minimum working you must write for a GDC-derived 3x3 system solution to get full marks?
- Only the final x, y, z values
- "GDC used to solve 3x3 system" plus 1 intermediate output
- Full manual elimination steps
- A graph of the system
Reveal answer
"GDC used to solve 3x3 system" plus 1 intermediate output —
IB markers only need confirmation you did not copy the answer from a third party source, no full manual steps required on Paper 2.

5. Common Pitfalls

Wrong move:

Writing only the final answer with no mention of GDC use

Why:

IB markers cannot award method marks if you do not show any evidence of your process

Correct move:

Add a 1-line note that GDC was used, plus one intermediate output value before your final solution

Wrong move:

Rounding polynomial coefficients to 2 s.f. before entering them into the GDC

Why:

Rounded inputs produce final outputs that fall outside the IB allowed tolerance range

Correct move:

Enter full unrounded values into your GDC, and only round the final root or system solution to 3 s.f.

Wrong move:

Trying to solve a 3x3 system manually on Paper 2

Why:

You waste 5+ minutes on a task explicitly designed for technology, leaving no time for later high-mark questions

Correct move:

Use the built-in GDC simultaneous solver for all 3-variable or larger systems

Wrong move:

Forgetting to adjust the graph window before using the root finder tool

Why:

Negative roots outside your default view range will be missed, costing you 1-2 accuracy marks

Correct move:

Set your x range to -10 to 10 before running the root finder for any polynomial

Wrong move:

Stopping after finding 2 roots for a cubic polynomial

Why:

You will miss the third real root and lose partial credit for incomplete solutions

Correct move:

Cross-reference the number of roots you found with the polynomial degree to confirm you have all real solutions

6. Quick Reference Cheatsheet

Task	GDC Step 1	GDC Step 2	Required Exam Annotation
2x2 Linear System	Open Simultaneous Equation Solver	Input 2 coefficients per equation	State 'GDC used for 2x2 system'
3x3 Linear System	Open 3-variable simultaneous tool	Enter all augmented matrix values	Write down 2 of the 3 output values
Quadratic Roots	Open Polynomial Solver, set degree 2	Input a, b, c coefficients	If there are no real roots, state 'no real solutions' (the graph has no x-intercepts)
Cubic / Quartic Roots	Graph over x=-10 to 10 range	Use root tool to capture all intercepts	List all real roots to 3 s.f.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · Paper 1
3x3 linear system solving
2024 · Paper 2
Cubic polynomial root finding
2023 · Paper 1
Linear-quadratic simultaneous solve

What's Next

Now that you can reliably use your GDC to solve systems and polynomial equations, you are ready to apply these skills to the real-world modelling problems that feature heavily on IB Math AI SL Paper 2. Mastering this technology workflow saves a great deal of algebraic manipulation time on most exam questions, freeing you up to focus on interpreting results and justifying conclusions, which carry high mark weightings on this course. You can also use these solving tools to cross-verify manual solutions and catch silly arithmetic errors that prevent many students from earning a 7 on their final exam.