Download N.6 - DPS ARE

Reasoning with Equations and Inequalities- A-REI
ELG.MA.HS.A.9: Solve equations and inequalities
in one variable.
 A-REI.4 Solve quadratic equations in one
variable.
 A-REI.4.a Use the method of completing the
square to transform any quadratic equation in
x into an equation of the form (x – p)2 = q that
has the same solutions. Derive the quadratic
formula from this form.
 A-REI.4.b Solve quadratic equations by
inspection (e.g., for x2 = 49), taking square
roots, completing the square, the quadratic
formula and factoring, as appropriate to the
initial form of the equation. Recognize when
the quadratic formula gives complex solutions
and write them as a ± bi for real numbers a and
b.
The Complex Number System – N-CN
ELG.MA.HS.N.6: Use complex numbers in
polynomial identities and equations.
 N-CN.C.8 (+) Extend polynomial identities to the
complex numbers. For example, rewrite x2 + 4 as
(x + 2i)(x – 2i).
 N-CN.C.9 (+) Know the Fundamental Theorem of
Algebra; show that it is true for quadratic
polynomials.
Algebra 2: The Complex Number System – N-CN
ELG.MA.HS.N.6: Use complex numbers in
polynomial identities and equations.
 N-CN.C.7 Solve quadratic equations with real
coefficients that have complex solutions.
Students will demonstrate command of the ELG by:
 Extending polynomial identities to the complex numbers.
 Showing that the Fundamental Theorem of Algebra is true for quadratic polynomials.
Vocabulary:
 Complex solution
 Fundamental Theorem of Algebra
 Polynomial identity
 Quadratic polynomial
Sample Assessment Questions:
1) Standard(s): N-CN.A.2 N-CN.B.5 N-CN.C.8
Source: https://www.illustrativemathematics.org/content-standards/HSN/CN/C/8/tasks/1659
Item Prompt:
For each odd positive integer n, the only real number solution to xn=1 is x=1 while for even positive integers n, x=1 and x=−1 are solutions to xn=1. In this problem we look for
all complex number solutions to xn=1 for some small values of n.
a.
Find all complex numbers a+bi whose cube is 1.
b.
Find all complex numbers a+bi whose fourth power is 1.
Correct Answer:
Solution: 2 Using Geometry (go to source link above)
Solution: 3 Solving Equations (go to source link above)