In this activity we seek to better understand the connection between roots and the plots of polynomials. We will restrict our attention to polynomials with real coefficients.

First, we need to be precise about the correct usage of some important language:

  • Expressions have values.
  • Equations have solutions: values of the variables that make the equation true.
  • Functions have zeros: input values that give output values of 0.
  • Polynomials (i.e., polynomial expressions) have roots.

These ideas are related, of course, as follows: A zero of a polynomial function, , is a root of the polynomial and a solution to the equation .

Please try to use this language correctly: Equations do not have zeros, and functions do not have solutions.

Give an example of a polynomial, and write a true sentence about related equations, functions, zeros, equations, and roots.
Sketch the plot of a quadratic polynomial with real coefficients that has:
(a)
Two real roots.
(b)
One repeated real root.
(c)
No real roots.

In each case, give an example of such a polynomial.

Can you have a quadratic polynomial with exactly one real root and complex root? Explain why or why not.
Sketch the plot of a cubic polynomial with real coefficients that has:
(a)
Three distinct real roots.
(b)
One real root and two complex roots.

In each case, give an example of such a polynomial.

Can you have a cubic polynomial with no real roots? Explain why or why not. What about two distinct real roots and one complex root?
For polynomials with real coefficients of degree to , classify exactly which types of roots can be found. For example, in our work above, we classified polynomials of degree and .
2024-10-10 13:42:13