CCSS.Math.Content.HSN-CN.C.9
The standard
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Common Core State Standards for Mathematics · The Complex Number System
What this standard means
Students need to understand that every polynomial has as many complex roots as its degree, when roots are counted with repeats. For quadratics, they should connect factoring, the quadratic formula, the discriminant, and complex conjugate roots.
Mastery looks like explaining why any quadratic has two complex solutions, even when the graph has no x-intercepts. Students often get stuck thinking “no real zeros” means “no solutions.” They may also forget repeated roots count twice, or miss that nonreal roots come in conjugate pairs when coefficients are real.
Ways to teach it
- Hands-on: Give students quadratic cards, have them sort by discriminant value, number of real roots, and number of complex roots.
- Prompt: Explain why x² + 4 has no real zeros but still has two complex solutions.
- Quick assessment: Ask students to solve three quadratics, one with two real roots, one repeated root, and one complex pair.
- Real-world connection: Use projectile height equations to show when real roots describe landing times and complex roots mean no real landing event.
Plan a lesson for CCSS.Math.Content.HSN-CN.C.9
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Related standards
- CCSS.Math.Content.HSN-CN.C.8
(+) Extend polynomial identities to the complex numbers.
- CCSS.Math.Content.HSA-APR.A.1
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- CCSS.Math.Content.HSG-SRT.D.10
(+) Prove the Laws of Sines and Cosines and use them to solve problems.
- CCSS.Math.Content.HSF-IF.C.8a
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