# Solving systems and polynomial equations with technology

> IB Mathematics Applications and Interpretation SL · IB AI SL
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u1-solving-systems-and-polynomial-equations/

We walk through step-by-step GDC workflows for solving linear systems and polynomial equations, plus IB-specific rules to earn full marks for technology-derived answers.

**Prerequisites:** [Basic polynomial terminology (degree, roots, coefficients)](https://www.owlsprep.com/study/ib-math-ai-sl-u1-intro-to-polynomials/); [Manual 2x2 linear system solving](https://www.owlsprep.com/study/ib-math-ai-sl-u1-manual-linear-systems/)

## Learning objectives

- Use a graphing display calculator (GDC) to solve 2x2 and 3x3 linear systems accurately
- Find all real roots of polynomial equations up to degree 4 using built-in GDC tools
- Verify algebraic solutions with technology to eliminate arithmetic errors
- Present GDC-derived working correctly to earn full method marks in IB exams

## GDC Pre-Configuration for IB AI SL Algebra Tasks

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.

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

*Notation:* N/A

*Example:* Inputting coefficients for $2x^2 + 5x - 3 = 0$ returns roots $x=0.5$ and $x=-3$

> **tip**
>
> Save your custom GDC settings to a profile so you can restore them in 10 seconds if your device resets mid-exam.

**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

**Check your understanding**

Confirm you can complete this setup on your own GDC:

1. 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.

   *Why:* All final answers in IB AI SL default to 3 significant figures unless the question explicitly states otherwise.

*Calculator:* allowed

## Solving 2x2 and 3x3 Linear Systems with GDC

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

**Exam command terms**

IB exam questions use specific command terms for system solving tasks:

- **Find** — You are permitted full GDC use, no manual working required beyond noting you used technology

- **Solve manually** — You must show full elimination/substitution steps, GDC only allowed for final verification

*Calculator:* allowed

## Finding Real Roots of Polynomial Equations Up to Degree 4

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.

> **warning**
>
> If your default graph window is set to only show positive x values, you will miss all negative roots for even and odd degree polynomials.

**Worked example:** Find all real roots of the cubic polynomial $f(x) = x^3 - 4x^2 + x + 6$ 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

*Calculator:* allowed

## Presenting GDC Working for Full Exam Marks

| 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 |

**Check your understanding**

Test your understanding of mark rules:

1. 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

   *Why:* IB markers only need confirmation you did not copy the answer from a third party source, no full manual steps required on Paper 2.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Writing only the final answer with no mention of GDC use
  - Why it fails: IB markers cannot award method marks if you do not show any evidence of your process
  - Correct: Add a 1-line note that GDC was used, plus one intermediate output value before your final solution
- **Wrong:** Rounding polynomial coefficients to 2 s.f. before entering them into the GDC
  - Why it fails: Rounded inputs produce final outputs that fall outside the IB allowed tolerance range
  - Correct: Enter full unrounded values into your GDC, and only round the final root or system solution to 3 s.f.
- **Wrong:** Trying to solve a 3x3 system manually on Paper 2
  - Why it fails: You waste 5+ minutes on a task explicitly designed for technology, leaving no time for later high-mark questions
  - Correct: Use the built-in GDC simultaneous solver for all 3-variable or larger systems
- **Wrong:** Forgetting to adjust the graph window before using the root finder tool
  - Why it fails: Negative roots outside your default view range will be missed, costing you 1-2 accuracy marks
  - Correct: Set your x range to -10 to 10 before running the root finder for any polynomial
- **Wrong:** Stopping after finding 2 roots for a cubic polynomial
  - Why it fails: You will miss the third real root and lose partial credit for incomplete solutions
  - Correct: Cross-reference the number of roots you found with the polynomial degree to confirm you have all real solutions

## 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. |

## 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.

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From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-ai-sl-u1-solving-systems-and-polynomial-equations/