Mathlets: Roots of Polynomials

Roots of Polynomials

Graphs a polynomial function P(x) with given real roots, showing the relationship between the roots and the coefficients of the polynomial.
How to use || Examples || Other Notes

How to use

The degree n of the polynomial P(x) (corresponding to the number of x-intercepts in the graph of y=P(x), counting multiplicity) can be changed using the "+" and "-" buttons under the label marked "n=". Each time the degree is changed, the values of the leading coefficient and roots are set to default values depending on the new degree.
The roots can be changed in either of two ways:
- Clicking and dragging the red dots on the graph.
- Entering values for any of the roots c₁ to c_n directly into the text input fields marked "c₁=", "c₂=", etc.
The leading coefficient a_n (which is not affected by the roots of the polynomial) can be changed by entering a value into the text input field marked "a_n=" (the displayed value of n changes with the degree of the polynomial).
Clicking the "Clear" button sets the values of n, c₁, c₂, etc., to default values (n=2, c₁=-1, c₂=1, other c_i blank) and redraws the graph.

The text input fields for the leading coefficient a_n and for the roots c₁ to c_n accept any decimal number. The buttons under the graph allow various manipulations of the graph coordinates.

Examples

Linear (degree 1): n=1, c₁=0, a₁=1 (multiplied out: y=P(x)=x)
Negative leading coefficient: n=2, c₁=-1, c₂=1, a₂=-1 (multiplied out: y=P(x)=1-x²)
Multiple root: n=3, c₁=-1, c₁=1, c₂=c₃=1 (multiplied out: y=P(x)= x³+x²-x-1)

Other Notes
The labels marked "a₀=", "a₁=", etc., show the values of the coeficients a₀, a₁, etc., of the polynomial P(x) when multiplied out. As the roots c_i change, the values of the coefficients a_i are updated automatically.

Multiple roots can be explored by making the values of two or more of the roots equal. Note however that irreducible quadratic factors of polynomials, which yield complex roots, are not supported by this applet.