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Expert: - 11/21/2010


Question

polynomial
The questions are based on polynomials & history of Maths.
Please see the attached picture.

Thanks for help.  

Answer
I will not deal with 1.  It is routine, but messy.

2. Is more interesting.  

A cubic polynomial always has one or three real roots.  

Therefore a parabola and the hyperbola xy = 1 will have 1,2, or 3

intersections.  (Yes, I said 1,2, or 3.)

Because:

xy = 1  -->  y = 1/x  and

y = Ax^2 + Bx + C  -->

1/x = Ax^2 + Bx + C  -->

Ax^3 + Bx^2 + Cx - 1 = 0, a cubic.

4x^3 - 6x^2 - 15x + 6 = 0  could come from this (in more than one

way, I would guess)

Such as:

4x^3 - 6x^2 - 15x = - 6

4x^2 - 6x - 15 = - 6/x

Now just write:

y = 4x^2 - 6x - 15  and  xy = - 6
...........................
In 2B, it's the same thing.  A quartic can have 0,2, or 4 real roots,

so a parabola and circle can have any number of intersections from 0

to 4.

(Yes, I said 0,1,2,3, or 4. 1 and 3 are possible.)

Multiply out:

(x - p)^2 + (x^2 - q)^2 = r^2

x^2 - 2xp + p^2 + x^4 - 2x^2q + q^2 = r^2

and you have a quartic.  To go back, try completing the square:

x^4 - 5x^2 - 2x + 6 = 0

x^4 - 6x^2 +     x^2 - 2x + 6 = 0

x^4 - 6x^2 + 9 + x^2 - 2x + 6 = 0 + 9

x^4 - 6x^2 + 9 + x^2 - 2x + 1 + 5 = 0 + 9

(x^2 - 3)^2 + (x - 1)^2  = 4

(y - 3)^2 + (x - 1)^2  = 4

and y = x^2

Cute.

(Yes, yes, you don't talk like that in the UK, I know.)

Paul Klarreich

Expertise

I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

Experience

I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.

Education/Credentials
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