Polynomials in one variable – graphs, values and roots
MathematicsAlgebraAges 12–13
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Algebra
Choose the degree and the coefficients of a polynomial P(x) of degree 1, 2 or 3; the simulation writes the polynomial in standard form, plots its graph and marks its roots (where the graph crosses the x-axis). Drag the point x along the x-axis to evaluate P(x) step by step and see that P(x) = 0 exactly at a root; move the coefficient sliders to watch the graph and its roots change. On the Division screen, divide P(x) by x − a or by a quadratic step by step with long or synthetic division, see that the remainder equals P(a) (remainder theorem) and is 0 exactly when x − a is a factor, and factorise cubics fully to find their roots. On the Factored form screen (degree up to 5), drag the zeros and set their multiplicity to see the graph cross (odd) or touch (even) the x-axis, read the end behaviour from the leading term, the y-intercept, complex zeros, a sign chart and the solution of a polynomial inequality.
Lesson: Polynomials in one variable; evaluating a polynomial; roots; polynomial division, remainder and factor theorems
What it shows
A polynomial in one variable is a sum of terms aₖxᵏ; its degree is the highest power with a non-zero coefficient, and a root is a value where P(x) = 0. A polynomial of degree d has at most d roots. Dividing P(x) by x − a gives a quotient Q(x) and a remainder R with P(x) = (x − a)Q(x) + R. The remainder theorem says R = P(a), and the factor theorem follows: x − a is a factor exactly when P(a) = 0. Dividing out one known root of a cubic leaves a quadratic that can be factorised to find the other roots. In factored form P(x) = a(x − r₁)^m₁…, each zero rᵢ is an x-intercept: the graph crosses the axis when the multiplicity is odd and touches it when it is even. The leading term axⁿ fixes the end behaviour, and a sign chart built from the signs of the factors solves P(x) > 0 or P(x) ≤ 0.
How to use
On the Graph and roots screen, set the Degree and the coefficients a₃ to a₀, then drag the point x along the x-axis to evaluate P(x). On the Division screen, choose the divisor x − a or x² + px + q and Long division or Synthetic division, then press Next step to build the working. Drag a on the graph, or click a tested candidate, until the remainder is 0, and read the full factorisation. On the Factored form screen, drag zeros along the x-axis, tap one and choose its Multiplicity, use Add zero or Remove zero, set a and the complex pair p ± qi, then pick an Inequality or the open-box Context.
Parameters you can change
- Degree of the polynomial Degree 1: a₁x + a₀, Degree 2: a₂x² + a₁x + a₀, Degree 3: a₃x³ + a₂x² + a₁x + a₀
- Coefficient a₃ (degree 3) -5–5
- Coefficient a₂ -5–5
- Coefficient a₁ -10–10
- Coefficient a₀ (constant term) -10–10
- Starting value of x -5–5
- Screen Graph and roots, Division, Factored form (zeros and multiplicity)
- Division method Long division, Synthetic division
- Divisor x − a, x² + px + q
- Value a in the divisor x − a -5–5
- Coefficient p of the divisor x² + px + q -5–5
- Coefficient q of the divisor x² + px + q -5–5
- Leading coefficient a (factored form) -3–3
- Real zeros (factored form), separated by ; – repeat a value for a multiple zero
- Real part p of the complex pair p ± qi -3–3
- Imaginary part q of the complex pair (0 = none) 0–3
- Inequality (factored form) P(x) > 0, P(x) ≥ 0, P(x) < 0, P(x) ≤ 0
- Context (factored form) None (all real x), Open-box volume, 0 < x < 4
- Show the factor lines
Questions to explore
- Why is x = 0 a root of P(x) exactly when the constant term a₀ equals 0?
- What is the smallest number of roots a polynomial of degree 3 can have, and why?
- Is x = −1 a root of x² − 2x − 3, and how do substitution and the graph both confirm it?
