Imaginary zeros of polynomials

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  1. Discussing the Relationship Between the Discriminant and X-intercepts on a Graph
    Sketch and compare the following quadratic functions:
    y=2x2+x15y=2{x^2} + x - 15
    y=2x2+x+15y=2{x^2} + x + 15
    1. Locating the Regions of Imaginary Zeros on Polynomial Graphs
      Indicate the region on the graphs where the non-real zeros occur.
      i) f(x)=110(x+2)(x1)(x3)(x5)f\left( x \right) = \frac{1}{{10}}\left( {x + 2} \right)\left( {x - 1} \right)\left( {x - 3} \right)\left( {x - 5} \right)
      ii) g(x)=110(x+2)(x1)(x28x+17)g\left( x \right) = \frac{1}{{10}}\left( {x + 2} \right)\left( {x - 1} \right)\left( {{x^2} - 8x + 17} \right)
      Topic Notes
      Based on the quadratic formula b±b24ac2a\frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} , non-real roots occur when the discriminant, b24ac{b^2} - 4ac , is negative. Non-real roots always occur in pairs.