The TI-84 can factor polynomials by finding their roots, which you then convert back into factored form. There’s no single “factor” button, but two reliable methods get you there: the built-in PlySmlt2 app for a fast algebraic approach, and the graphing method for a more visual one. Both work on the TI-84 Plus and TI-84 Plus CE.
Why the TI-84 Finds Roots Instead of Factors
The TI-84 doesn’t output a neatly factored expression like (x + 3)(x − 2). Instead, it finds the zeros (roots) of a polynomial, which are the x-values where the expression equals zero. Once you have those roots, converting them into factors is straightforward. If a root is x = 5, the corresponding factor is (x − 5). If a root is x = −3, the factor is (x + 3). You’re essentially reversing the zero-product property from algebra.
Method 1: The PlySmlt2 App
This is the fastest approach and works entirely without graphing. The app is called the Polynomial Root Finder and Simultaneous Equation Solver, and it comes preloaded on TI-84 Plus and TI-84 Plus CE calculators.
Here’s the full sequence:
- Open the app. Press the [APPS] key and scroll down to PlySmlt2 (it’s usually option 9, but the position may vary). Press [ENTER] to launch it.
- Select Polynomial Root Finder. Choose Option 1 from the app’s main menu.
- Set the degree. The app asks for the ORDER of your polynomial. For a standard quadratic like 2x² + 5x − 3, the order is 2. For a cubic, choose 3.
- Enter your coefficients. Type each coefficient in order from highest degree to lowest, using the arrow keys and [ENTER] to move between fields. For 2x² + 5x − 3, you’d enter 2, then 5, then −3.
- Solve. Press the [GRAPH] key, which doubles as the SOLVE button inside this app. The calculator displays the roots.
To exit, press [2nd] [QUIT] to move between screens, or choose option 6 (QUIT APP) to close the program entirely.
Reading the Results
The app gives you root values. If it shows x₁ = 0.5 and x₂ = −3, those are your zeros. To write the factored form, reverse each root into a factor. A root of 0.5 means x = 1/2, so the factor is (2x − 1) after clearing the fraction. A root of −3 gives (x + 3). The full factored form would be (2x − 1)(x + 3).
If the roots are complex numbers (involving i), the polynomial doesn’t factor over the real numbers. The app will show you this clearly.
Method 2: Graphing and Finding Zeros
If you prefer a visual approach, or if PlySmlt2 isn’t available on your calculator, you can graph the polynomial and locate where it crosses the x-axis. Each crossing point is a root.
- Enter the function. Press [Y=] and type your polynomial into Y1. For example, type 2X² + 5X − 3.
- Graph it. Press [GRAPH]. You should see the curve on screen. If the curve doesn’t appear or looks cut off, press [ZOOM] and select 6:ZStandard to reset the viewing window, or use [WINDOW] to adjust the range manually.
- Open the zero finder. Press [2nd] [TRACE] (which opens the CALC menu), then select option 2: zero.
- Set the left bound. The calculator asks “Left Bound?” Use the left arrow key to move the cursor to a point clearly to the left of the x-intercept you want to find, then press [ENTER].
- Set the right bound. When prompted for “Right Bound?”, move the cursor to the right of that same intercept and press [ENTER].
- Make a guess. When it says “Guess?”, move the cursor close to where the curve crosses the axis and press [ENTER]. The calculator displays the zero at the bottom of the screen.
Repeat this process for each x-intercept. A quadratic has at most two, a cubic has at most three, and so on. If the curve touches the x-axis without crossing it, that root has an even multiplicity, meaning the factor appears twice. For instance, if x = 4 is a “touch” point, the factored form includes (x − 4)².
Converting Decimal Roots to Clean Factors
Both methods sometimes give roots as decimals like 0.333333 or 1.5, which are harder to work with when writing factors. You can force the TI-84 to display fractions instead.
On the home screen after finding a root, press [MATH] and select option 1: ►Frac, then press [ENTER]. This converts the last answer to a fraction. So 0.333333 becomes 1/3, and 1.5 becomes 3/2.
You can also set the calculator to show fractions automatically. Press [MODE], arrow up to the “Answers:” row, then arrow right to highlight FRAC-APPROX and press [ENTER]. From that point, results display as fractions whenever possible.
Turning Fractional Roots Into Integer Factors
A fractional root like x = 3/2 doesn’t directly translate to a neat factor of (x − 3/2). To clear the fraction, multiply through by the denominator. The factor becomes (2x − 3). Here’s the general rule: if a root is p/q in lowest terms, the factor is (qx − p).
For example, say PlySmlt2 gives roots of −1/4 and 2 for the polynomial 4x² − 7x − 2. The root −1/4 becomes the factor (4x + 1), and the root 2 becomes (x − 2). The factored form is (4x + 1)(x − 2). You can verify by expanding this back out on the home screen.
Factoring Higher-Degree Polynomials
Both methods extend beyond quadratics. In PlySmlt2, set the order to 3 for a cubic, 4 for a quartic, and so on. The app handles polynomials up to degree 10. When graphing, you simply repeat the zero-finding process for each x-intercept.
For cubics and higher, keep in mind that some roots may be complex (non-real). PlySmlt2 will display these, but the graphing method won’t, since complex roots don’t appear as x-intercepts. If your cubic has only one visible crossing on the graph, the other two roots are complex conjugates, and the polynomial factors into one linear factor and one irreducible quadratic.
Quick Example: Factor 6x² + x − 12
Open PlySmlt2, choose Polynomial Root Finder, set order to 2, and enter coefficients 6, 1, −12. Press SOLVE. The calculator returns x = 4/3 and x = −3/2. Convert these to factors: 4/3 gives (3x − 4), and −3/2 gives (2x + 3). The factored form is (3x − 4)(2x + 3). Multiply it out to confirm: 6x² + 9x − 8x − 12 = 6x² + x − 12.