Statement and Implications

Algebra 2 · Axiom Academy

LESSON The Fundamental Theorem of Algebra Every degree- n polynomial ( ) has exactly n roots in , counted with multiplicity — so every equation can be solved. The theorem comes in two equivalent forms. The existence form says every non-constant polynomial with complex coefficients has at least one root in . Apply it repeatedly and you reach the counting form : a polynomial of degree has exactly n roots in , counted with multiplicity. Existence: at least one root lives in ℂ Counting: exactly n roots in ℂ, with multiplicity Exactly n roots only works if a repeated root is counted as many times as it appears. Watch a quadratic s two roots slide together: at the instant they meet there is one location but still two roots — a double root of multiplicity 2. Two roots become one — but still count twice At k = 3 , x^2 - 4x + 3 = (x-1)(x-3) has two distinct roots, 1 and 3 . At k = 4 , x^2 - 4x + 4 = (x-2)^2 has the single location 2 — but with multiplicity 2, so it still counts as two roots in . The tally never drops below the degree. 3. Real Coefficients: Roots in Conjugate Pairs When the coefficients are real , the non-real roots are forced to pair up. If a + bi is a root (with ), then its mirror image a - bi is a root too — the two are complex conjugates, reflections of each other across the real axis. This one implication has real bite — but only for real coefficients: The non-real roots always come in pairs, so their count is even .

This is the written version of the interactive lesson above. See the full Algebra 2 course.