Discussion and Homework
Thirteenth Reading Assignment

Notes for the Readings

Fermat uses notation similar to Viète. Read x for a and y for e.

Questions for Discussion

Reading 62

  1. Find a precalculus or college algebra textbook and write down the following to bring to class for discussion:
    a. The Remainder Theorem
    b. The Factor Theorem
    c. The Fundamental Theorem of Algebra
    d. (Descartes) Rule of Signs
  2. Can you find all of the items in number 1 in the reading?

Reading 63

  1. Viète was concerned with homogeneity of terms, that is you must have something like A3 = BpA. Descartes is not concerned with homogeneity. How does he "explain" this?
  2. Verify that BE = BD × BC and BC = BE ÷ BD in Figure 63.1.
  3. Verify that GI = GH1/2 in Figure 63.2 (use the Pythagorean Theorem).
  4. Verify that z = OM = 1/2 a + (1/4 aa + bb)1/2 in Figure 63.3.
  5. Solve z2 = az + bb using the quadratic formula. In Figure 63.3, what is the other root, namely 1/2 a - (1/4 aa + bb)1/2.
  6. Read carefully Descartes' Method of Obtaining the Equation of a Curve. We will go through and verify the steps.

From The Geometry of Rene Descartes

  1. If z and y are two mean proportionals between a and q, what does that mean algebraically (in terms of ratios)?
  2. If z is one of the mean proportionals, why is it "evident" that a : z = z : z2/a = z2/a : z3/a2?
  3. Using question 1, why must z3 = a2?
  4. What is the latus rectum of a parabola? Where does it appear in the equation of a parabola. Look it up!
  5. Follow carefully through the footnote [231] in the reading. We will go through the verification carefully.

"On Analytic Geometry"

  1. Where is the idea of coordinate axes first mentioned?
  2. Fermat takes the reader through the conic sections. Do you recognize the modern equations of each?
  3. Sketch the graph of xy = c. Using all that you know about inversions and shifting, sketch the graph of (x - 2)(3 - y) = c - 6. Does it look like the second picture on page 391?
  4. On the top of page 392, why is it "easy to show" that
    (NO2 + NO.OR)/OR2 = (NZ2 + NZ.ZI)/ZI2?
  5. Show that (b2 - a2)/e2 = a given ratio represents the equation of an ellipse.
  6. Show that (a2 + b2)/e2 = a given ratio represents the equation of a hyperbola.

Homework Problems

  1. Use Descartes rule of signs to determine the number of possible positive and the number of possible negative real zeroes of the polynomial x4 - 5x3 + 5x2 + 5x - 6.
  2. In the Cardano/Viète homework you reduced a cubic to a depressed cubic and a quadratic to a depressed quadratic. Elsewhere in La Géométrie of Descartes (page 376) he states a general rule : (assuming the coefficient of the highest term is 1) "First, we can always remove the second term of an equation by diminishing its true roots by the known quantity of the second term divided by the number of dimensions of the first term, if these two terms have opposite signs; or, if they have like signs, by incresing the roots by the same quantity. What would the substitution be to reduce the following to their depressed forms?
    a. ax4 + bx3 + cx2 + dx + e
    b. ax5 + bx4 + cx3 + dx2 + ex + f
    c. ax6 + bx5 + cx4 + dx3 + ex2 + fx + g
  3. Elsewhere in La Géométrie of Descartes (page 379) tells how to "multiply or divide all the roots of an equation by a given quantity, without first determining their values." His example involves transforming x3 - 31/2x2 + 26/27 x - 8/(27 31/2) = 0. Descartes says "multiply the second term by the square root of 3, the third by 3, and the last by three square roots of 3." He ends up with y3 - 3y2 + 26/9 x - 8/9 = 0. What substitution is Descartes really making here?
  4. Verify like we did in class for the others that MG is the solution of z2 = az - bb in Figure 63.4.
  5. In Reading 63 (page 339) Descartes states that yy = cy - cx/b y + ay -ac is a hyperbola. Let a = b = c = 1. Use rotation of axes to verify that this equation is indeed a hyperbola. Sketch the graph. Note: Your calculus book will have a section on rotation of axes— look in the index or the table of contents. If you had linear algebra, you could use those techniques instead.
  6. In the Fermat reading (page 395), he states that "The most difficult type of equation is that containing, along with a2 and e2, terms involving ae, or other given magnitudes, etc." These problems involve rotation of axes (as you saw in problem 3). Fermat's example is b2 - 2a2 = 2ae + e2. Find a theorem in your calculus book that uses coefficients of the equation to determine the type of conic section you have. State the theorem and verify that Fermat's example does give an ellipse as he says.

A pdf version of this page and the accompanying reading assignment.

Go back to course home page
Go back to my home page.