Identify zeros from factorizations and use them to sketch polynomial graphs.
Distinguish real and complex zeros in a factorization for graph sketching
Problem
For \((x~-~2)(x^2~+~1)\), identify the real zeros, complex zeros, and \(x\text{-}\text{intercepts}\) of the real graph.
Big Picture
What this problem is really about
Separate algebraic zeros from visible x-intercepts. Solve each factor over the complex numbers, classify the resulting zeros as real or nonreal, and remember that only real zeros can appear as points where the graph meets the x-axis. The nonreal zeros still count toward the polynomial’s degree even though they do not appear on the real graph.
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