Solving systems and polynomial equations with technology

IB Mathematics Applications and Interpretation SL· SL 1.8: Technology for systems of linear equations in up to three variables and polynomial equations· 12 min read

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

✓ Calculator OK

Keep full calculator precision, and give final answers exactly or to three significant figures unless another accuracy is requested. Set the window to fit the function: no fixed window guarantees every real root is visible.

📘 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

Check the graph window before solving $x^{2} - 400 = 0$ .

1
Enter the polynomial coefficients $1, 0, - 400$ in the solver. The exact roots are $x = - 20$ and $x = 20$ .
2
For a graphical check, use a window such as $- 25 \leq x \leq 25$ and $- 450 \leq y \leq 250$ . Both roots are outside the default $[- 10, 10]$ x-window.

✓ Quick check

Confirm you can complete this setup on your own GDC:

How should final numerical answers be presented?
- Exact or 3 significant figures, unless otherwise specified
- Always 3 decimal places
- Round every intermediate value to 3 significant figures
Reveal answer
Exact or 3 significant figures, unless otherwise specified —
Keep full calculator precision during calculations. Give final answers exactly or to three significant figures unless another accuracy is requested.

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

✓ Calculator OK

A GDC solves systems of up to three variables on both AI SL papers. Enter the coefficients and constants carefully, label every requested variable, and substitute the solution back to check it. SL 1.8 prescribes no particular method.

\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 (solve-system) function on your GDC and select 3 variables
2
Input each equation's coefficients exactly as written, including the right-hand-side value
3
Run the solver to get $x = 1$ , $y = 0$ , $z = 1$
4
Substitute 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

✓ Calculator OK

Manual factorization is almost never expected in IB AI SL for degree 3 or higher. Use the GDC polynomial solver or graph root finder 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 and set the degree to 3
2
Input the coefficients a=1, b=-4, c=1, d=6
3
Run the solver to get $x = - 1$ , $x = 2$ , $x = 3$
4
Graph the function to confirm all 3 x-intercepts appear in your window

4. Communicating Mathematical Working with a GDC★★★★☆⏱ 8 min

✓ Calculator OK

Component	What to show
Solutions	Label all requested values; give exact values or the required accuracy.
Mathematical working	Show equations, models, reasoning or checks relevant to the question.
Use of technology	Writing 'GDC used' alone is not mathematical working and does not guarantee marks.

✓ Quick check

Test your understanding of mark rules:

How should a solution be recorded when x, y and z are all requested?
- Label all three requested solution values and show any setup or reasoning the question requires
- Write only that a GDC was used
- Give two of the three variables
Reveal answer
Label all three requested solution values and show any setup or reasoning the question requires —
Marks depend on the particular question and markscheme, not a fixed calculator note or number of intermediate outputs.

5. Common Pitfalls

Wrong move:

Treating a calculator-use note as a substitute for mathematics

Why:

The note shows none of the equations, answers or reasoning the question asks for.

Correct move:

Label every requested value and show the setup, reasoning or check the question requires. A calculator-use note alone does not guarantee marks.

Wrong move:

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

Why:

Rounding the inputs can change the roots or solutions.

Correct move:

Keep full calculator precision during calculations; give the final answer exactly or to three significant figures unless another accuracy is requested.

Wrong move:

Assuming every cubic has three distinct real roots, or reading roots off an unadjusted window

Why:

A cubic has one, two (with a repeated root) or three distinct real roots, and roots outside the displayed window are easy to miss.

Correct move:

Enter all coefficients, including zero coefficients, in the polynomial solver, then check on a window that fits the function. A repeated root may touch the x-axis without crossing it, and no fixed window guarantees every root is visible.

6. Quick Reference Cheatsheet

Task	GDC Step 1	GDC Step 2	Written answer
2x2 Linear System	Open Simultaneous Equation Solver	Enter all coefficients and the right-hand-side constant	Label all requested variables
3x3 Linear System	Open 3-variable simultaneous tool	Enter all augmented matrix values	Label all requested variables
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	Enter all coefficients, including zero coefficients, in the polynomial solver	Check using an adjustable window; repeated roots may touch the axis	Exact or 3 significant figures, unless otherwise specified

What's Next

Next, apply these GDC solving tools to the real-world modelling problems that feature heavily on AI SL Paper 2, where interpreting and justifying the results carry high mark weightings.