Challenge Activity

This activity will help you compare modern day techniques to solving roots of polynomials with that of Descartes’ method. 

How it connects to common core curriculum:

Standard A.APR.3

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

 

1.      Remind yourself of the Rational Zeros Theorem here. 

2.      Remind yourself of Desecrates’ Rule of Signs here and here.

3.      Use the online calculator, desmos.com/calculator to graph the polynomials. 

4.      Remind yourself of synthetic division here.

5.      Remind yourself of factoring polynomials here. 

 

Use these techniques that we use today to help you find the roots of the polynomials below:

1.

2. =0

3.

(Analyzing and Solving Polynomial Equations)

 

 

Then use Descartes’ methods (Quadratic, Cubic, Quartic, Quintic, and Sextic) and these applets to solve:

1.

2. =0

3.

(Analyzing and Solving Polynomial Equations)

Quadratics Applet

Use the applet below to see the x-values of the intersections of the circle and the line. Compare with the x-values of the roots of f(x).

Move p (the coefficient on x in f(x)) and see how it effects the circle.
Move q (the constant of f(x)) and see how it moves the line.
Click the checkbox on f(x) to see the quadradic and its roots.

Quartics Applet

Use the applet below to see the x-values of the intersections of the circle and the parabola. Compare with the x-values of the roots of g(x).

Move p (the coefficient on x^2 in g(x)) and see how it effects the circle.
Move q (the coeffiecient on x in g(x)) and see how it effects the circle.
Move r (the constant of g(x)) and see how it effects the circle.
Click the checkbox on g(x) to see the quartic and its roots.

Sextics Applet

Visit this website.

 

Did you get the same answer both times?

 

 

Which one gave a more exact answer?

 

 

Which one do you prefer?