Experience First

The students will see in the first problem that it does not cross the x-axis in the coordinate plane.  Normally, this would lead them to say “No Solution” since there are no x-intercepts.  However, we know that there should be two solutions because of the degree of the polynomial.  We introduce the imaginary number, i, and give the students a chance to continue solving for all solutions.

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They may struggle with how to write their responses, so walk around the room and show them that i(2) can be written as 2i.  If they have trouble going from zeros to factored form in #4, have them write each zero separately so they can see how to move everything to the left.  Some may not know that                , so have them consider what.                 would be.

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Formalize Later

Pose the question “how can you tell we’ll have imaginary solutions from looking at the equation” to get the students thinking about the connection between square rooting a negative number and getting an imaginary number.  Show how the complex part of the solutions cancel each other out when multiplied together, producing the original polynomial with all real coefficients.  This will help explain the idea of a conjugate pair. Solving with a square root is a great way to show how you always have an even number of imaginary OR irrational solutions for a polynomial since you have to consider the + and -.

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Add the note about adding and subtracting complex numbers during the Check Your Understanding.  It is important for questions on the SAT or ACT like #4.