**Title:** Who am I? Find A Polynomial From Its Roots

**Description:**
Students analyze the roots and end behavior of a polynomials and write the equation of a polynomial under given conditions. Students apply theorems concerning the multiplicity of roots, conjugates of irrational or complex imaginary roots to find a polynomial under given conditions. Students will factor polynomials to find the complex roots. Students graph polynomials and determine the local extrema. This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project funded by the Malone Family Foundation.

**Standard(s): **

[TC2] CA2 (9-12) 9: Practice ethical and legal use of technology systems and digital content.

[TC2] CA2 (9-12) 11: Critique digital content for validity, accuracy, bias, currency, and relevance.

[MA2013] AL2 (9-12) 5: (+) Extend polynomial identities to the complex numbers.

[MA2013] AL2 (9-12) 6: (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. [N-CN9]

[MA2013] AL2 (9-12) 16: Know and apply the Remainder Theorem: For a polynomial *p*(*x*) and a number *a*, the remainder on division by *x* - *a* is *p*(*a*), so *p*(*a*) = 0 if and only if (*x* – *a*) is a factor of *p*(*x*). [A-APR2]

[MA2013] AL2 (9-12) 17: 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. [A-APR3]

[MA2013] AL2 (9-12) 30: Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [F-IF7]

[MA2013] ALT (9-12) 5: (+) Extend polynomial identities to the complex numbers. [N-CN8]

[MA2013] ALT (9-12) 6: (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. [N-CN9]

[MA2013] ALT (9-12) 16: Know and apply the Remainder Theorem: For a polynomial *p*(*x*) and a number *a*, the remainder on division by *x* - *a* is *p*(*a*), so *p*(*a*) = 0 if and only if (*x* – *a*) is a factor of *p*(*x*). [A-APR2]

[MA2013] ALT (9-12) 17: 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. [A-APR3]

[MA2013] ALT (9-12) 30: Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [F-IF7]

[MA2013] AM1 (9-12) 11: (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Understand the importance of using complex numbers in graphing functions on the Cartesian or complex plane. [N-CN9] (Alabama)