Hints^243
- Look at x2 - y2 and xy.
- The local extrema are those k for which y = k is tangeut to the graph
of the cubic. - 2(ab - cd) = [u” + b2 - c2 - d2] - [(Q - b)2 - (c - d)2]. Factor the
difference of squares. - All the coefficients must be rational so we can assume that they are
integers. Reduce to the manic case with prime constant coefficient,
so that a great deal can be said about the rational zeros. Recall the
role of the Intermediate Value Theorem in guaranteeing real zeros. - Let u = a/(bc - a2), etc. and look at VW - u2, etc.
- Explore the situation for small n and make a conjecture.
- If r is a zero of f(t), what other zeros must there be?
- Apply the AGM inequality to u2 + 3x2.
- Factor z3 + y3 + .r3 - 3xyz.
- Every zero of the divisor should make the dividend vanish.
- The equation is (2 + 1)3 - x3 = y”. Express as a quadratic in x and
complete the square. - Take the differences of the equations in pairs.
51. Sketch the graphs of y = 6x2, y = 77x - 147 and y = 77[x] - 147.
53. Determine the form of a cubic whose zeros are u = 49-a, etc. Heron’s
formula for the area will be useful.
56. The denominator is a difference of squares. Use the Factor Theorem.