multiplicity

Problem: Find all zeros and their multiplicities for the polynomial f(x) = 2x⁴ − 12x³ + 18x², and describe how the graph behaves at each zero. Step 1: The polynomial is already in factored form. Read off each zero directly from the factors. A simple zero means the factor (x−r)(x - r)(x−r) appears exactly once; the graph crosses the x-axis at a nonzero angle. A repeated zero means the factor appears two or more times. At a double root (multiplicity 2), the graph merely touches the axis. At a triple root (multiplicity 3), the graph crosses but with a distinctive flattening. In general, even multiplicity → bounce, odd multiplicity → cross. Multiplicity connects algebra and graphing in a powerful way: knowing a zero's multiplicity tells you immediately whether the graph crosses or bounces at that intercept. It also underpins the Fundamental Theorem of Algebra, which guarantees that a degree-nnn polynomial has exactly nnn zeros when counted with multiplicity (over the complex numbers). In calculus, a zero of multiplicity m≥2m ≥ 2m≥2 is also a zero of the derivative, which matters when analyzing critical points and curve behavior. Mistake: Forgetting that the sum of all multiplicities must equal the degree of the polynomial. Correction: Always verify by adding the multiplicities together. If their sum doesn't match the polynomial's degree, you have either missed a factor or miscounted a repeated one. Mistake: Confusing the behavior at even vs. odd multiplicity zeros — thinking the graph always crosses at every x-intercept. Correction: The graph only crosses at zeros with odd multiplicity. At zeros with even multiplicity, the graph touches the axis and turns back. Sketch a quick sign chart or test point if you're unsure.