Advanced High-School Mathematics

SECTION 3.6 Cubic Discriminant 173

1 2 3

2

4

6

x

y

x^3 −xy+2=0

y+x =c

2

level curves

Equation 1: y=2/x+x²

Equation 2: y=c−x²

Equation 3: y=2.1−x²

4. Compute the maximum value ofEquation 4: y=4.2−x^2 xy, given that
   x^2 +y= 4.


5. The polynomialS(x 1 ,x 2 ,x 3 ) =x^31 +x^32 +x^33 is symmetric inx 1 ,x 2 ,x 3
   and can be expanded in the elementary symmetric polynomials

σ 1 =x 1 +x 2 +x 3 , σ 2 =x 1 x 2 +x 1 x 3 +x 2 x 3 , σ 3 =x 1 x 2 x 3.

Watch this:

x^31 +x^32 +x^33 = (x 1 +x 2 +x 3 )^3

− 3 x^21 (x 2 +x 3 )− 3 x^22 (x 1 +x 3 )− 3 x^23 (x 1 +x 2 )− 6 x 1 x 2 x 3

= (x 1 +x 2 +x 3 )^3 −3(x 1 +x 2 +x 3 )(x 1 x 2 +x 1 x 3 +x 2 x 3 )

+ 3x 1 x 2 x 3

= σ^31 − 2 σ 1 σ 2 + 3σ 3.

Now try to write the symmetric polynomial x^41 +x^42 +x^43 as a

polynomial inσ 1 , σ 2 , σ 3.

6. Let f(x) = ax^3 +bx^2 +cx+d, and so the derivative is f′(x) =
   3 ax^2 + 2bx+c. Denote byR(f) the determinant