By Jack Gerard, Updated Aug 30, 2022
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What Is a Polynomial?
A polynomial is an algebraic expression that contains one or more powers of a single variable (typically x) raised to positive integer exponents. It can be written in the standard form:
—y = a_nx^n + a_{n-1}x^{n-1} + … + a_1x + a_0
Each coefficient a_i can be any real number, and the highest exponent n determines the degree of the polynomial. A true polynomial involves only one variable; if multiple variables appear, the expression is not a polynomial.
Solving Simple Polynomials with the Equation Solver
The TI‑84 Plus includes an Equation Solver that works best for linear equations (degree 1) or any single‑variable equation that can be set to zero. To use it:
- Press the MATH button, then select 0:Solver… (or simply press the 0 key).
- Enter your equation, making sure the left‑hand side equals zero. For example, type 2x+1=0 or simply 2x+1 if the calculator automatically moves the constant to the right.
- Press ENTER and then provide a reasonable lower and upper bound for x when prompted.
- Press ENTER again. The calculator will iterate within the bounds and return the root that satisfies the equation.
Because the Equation Solver handles only one variable, it is ideal for equations like 2x+1=0 or any linear expression that reduces to that form.
Solving Higher‑Degree Polynomials with the Poly Root Finder
For polynomials of degree 2 or higher, the Poly Root Finder—part of the PolySmlt application—provides a robust solution. Follow these steps:
- Press the APPS button and scroll to the PolySmlt entry. The hotkeys only cover the first ten items, so you may need to use the DOWN arrow several times.
- Press ENTER to launch the application, then select 1:Poly Root Finder from the submenu.
- When prompted, enter the degree of the polynomial (the highest exponent). For example, for a quadratic use "2".
- Input the coefficients for each term, starting with the highest‑degree term. If your polynomial is 3x^3-2x^2+4x-5=0, enter 3, -2, 4, and -5 in that order.
- Press the GRAPH key (located beneath the SOLVE button) to start the calculation.
- After a moment, the calculator will display all real roots found. Any entry that yields no real solution will be labeled NONREAL.
This tool works for polynomials up to the calculator’s degree limit (typically 10). It is a powerful way to find all real solutions without manual factoring.
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