CCSS.Math.Content.HSN-CN.C

CCSS.Math.Content.HSN-CN.C is a cluster heading. These are the individual standards under it.

• CCSS.Math.Content.HSN-CN.C.7

  Solve quadratic equations with real coefficients that have complex solutions.

• CCSS.Math.Content.HSN-CN.C.8

  (+) Extend polynomial identities to the complex numbers.

• CCSS.Math.Content.HSN-CN.C.9

  (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

01

What Students Need to Do

Students solve polynomial equations when some roots are complex. They use complex arithmetic to rewrite and verify polynomial identities. They also connect roots to linear factors.

02

What Mastery Looks Like

Students solve polynomial equations with real or complex roots and simplify powers of i correctly. They connect each root to a linear factor and verify identities by expansion. They recognize conjugate root pairs when coefficients are real.

03

Common Misconceptions

Students may treat i² as 1 instead of -1, or write √-9 as -3. They may forget that nonreal roots of polynomials with real coefficients come in conjugate pairs. They may find roots correctly but write the wrong signs in the matching factors.

04

How to Assess It

Give this exit ticket: Solve x² + 4x + 13 = 0, write the polynomial as linear factors, and verify by multiplying.

1. 01

   Match cards showing a polynomial, its complex roots, and its linear factorization, then confirm each set by multiplication.

2. 02

   Discuss and write a response to: “Why must nonreal roots of a real-coefficient polynomial occur in conjugate pairs?”

3. 03

   Run a root-factor relay where teams solve one polynomial, write its factors, and check another team’s multiplication.

4. 04

   Solve -5t² + 20t = 25 for an unreachable projectile height, then explain why its complex times have no physical meaning.

• CCSS.Math.Content.HSA-APR.C.4

  Prove polynomial identities and use them to describe numerical relationships.

• CCSS.Math.Content.HSA-APR.C

  Use polynomial identities to solve problems

• CCSS.Math.Content.HSN-CN.A

  Perform arithmetic operations with complex numbers.