Algebra II Task: Exploring Polynomials

Essential and guiding questions:

How are these coefficients related to the zeroes of the polynomial?
Does it matter what form the zeroes take (positive, negative, fractions, irrational numbers, or complex numbers)? Why or why not?
Why do these patterns rely on the leading coefficient being 1? How are your patterns affected if the leading coefficient is not 1?

Blooms taxonomy level:

Understanding

Differentiation suggestions:

If students can’t get started….

Try a simpler case. What would happen if you only used r and s? How would you “build” your factors? How would you multiply these together?

Extension suggestions:

If students finish early….

What would happen if you chose 4 zeroes? What form would your polynomial take if you multiplied the four factors? How do your zeroes relate to the coefficients of your fourth‐degree polynomial?
Would your patterns still work if the leading coefficient of your third‐degree polynomial was not 1? Could you adjust your patterns in this case?
Will your patterns work no matter what form your zeroes take (positive, negative, fractions, irrational numbers, or complex numbers)?

Originator: