a) If there are four real zeroes, then how many complex zeroes are there? What sort of factor can generate two complex (or, to make things simpler, two imaginary) zeroes? (It may be helpful to use "x<sup>2</sup>" as your leading term, rather than something like "6x<sup>2</sup>", to make later steps simpler.)
b) If x = -5 is a zero three times, then what factor occurs three times? (It may be helpful to use "x" as your leading term, rather than something like "25x", to make later steps simpler.)
From (a) and (b), you get five of the six zeroes taken care of. Some care needs to be taken in finding the final zero, because there are other considerations.
c) For the function, as x gets very large, to behave roughly like y = -4x<sup>6</sup>, the leading term must be -4x<sup>6</sup>. So you're going to need to multiply your five factors above by something that gives a leading coefficient of -4x<sup>6</sup>.
d) To get a y-intercept at y = 5, your sixth factor will need to be of the form "x + m", where "m" is some number which, when multiplied against the constant term of the product of the other factors, gives a result of "5" for the new constant. So multiply all the constant terms of your factors from (a) and (b), and multiply by your number from (c). Multiply this by "m", set equal to "5", and solve for whatever m needs to be.
e) Graph.
Hope that helps a bit. If you get stuck, please reply showing how far you've gotten. Thank you.
Eliz.
