Experience First

In this lesson students use three clues to help them describe the zeros of a polynomial function, incorporating their knowledge from several previous lessons. Students must use their logical reasoning skills to determine how many real and imaginary zeros the polynomial will have as well as the multiplicity of the real zero (since f(x) is always greater than or equal to zero, the multiplicity must be even since the graph must bounce off the x-axis and stay positive at x=2!) They will review how to determine the total number of zeros of a polynomial and the idea that complex zeros come in conjugate pairs. 

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Although not many new ideas are introduced in this lesson, we find that students need days like this to consolidate their understanding and put the pieces together.

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Formalize Later

Decide if you want students to factor completely in question 4 (so only linear factors of the form (x-c) remain, where c is a complex number), or if you will accept (x^2+2x+10) as one of the factors. Either way students will have to find all the real and imaginary zeros in question 3. It may be worth having the discussion about when and why you may see a quadratic expression in factored form, namely when the factor is irreducible, meaning it can not be broken down further over the real numbers. 

If students factor completely, emphasize the importance of parentheses when writing a factor that includes a complex zero.