Cubic functions.....?

Question

Could somebody please give me an example of a cubic function that contains one real, and two complex numbers? Two examples would be even nicer - Thanks

Answer 1

I assume you mean a cubic function with one real and two complex roots.

First find a quadratic with complex roots.

As an example let the roots be 3+4i and 3-4i
Sum of roots = 6
Product of roots = 25

Quadratic
x^2 - 6x + 25 = 0

Multiply this by (x-1)

(x-1)(x^2-6x+25)=0
x^3 - 7x^2 + 31x - 25 = 0

You can use a similar method to produce a second cubic equation with one real to two complex roots.

A second example is 2x^3 - 21x^2 + 348x -169 =0
(Hint: the real root is 0.5)

Answer 2

The complex zeros (which I assume you mean here) come in conjugate pairs, so just construct:

x = ±2i, so x² = -2, x² + 2 = 0,
and let x = -3, so x + 3 = 0. Then
(x² + 2)(x + 3) = x^3 + 3x² + 2x + 6 = 0 is a cubic polynomial and

f(x) = x^3 + 3x² + 2x + 6 is a cubic function with 1 real zero and 2 complex zeros.

Without bothering to start all the way from scratch,

let f(x) = (x² + 5)(x - 3) = x^3 - 3x² + 5x - 15, and f(x) will have ±5i and 3 as zeros.

Answer 3

You can invent plenty of examples for yourself, even if you don't know anything about complex numbers!!!
Choose a real root, and the factor corresponding to that. e.g. Suppose you want the real root to be -2. Then use x+2 as a factor.

Now, for the other factor, use a quadratic expression that hasn't real roots, e.g. x^2 + positive number (x^2 + 1, x^2 + 6, anything like that), or a 3 term quadratic in which b^2 is less than 4ac,
e.g. x^2 + 4x + 5

Multiply these two factors together, and you have the sort of cubic you asked for.

e.g. (x + 2)(x^2 + 6)
= x^3 + 2x^2 + 6x + 6

Another: (x - 1)(x^2 + 4x + 5)
= x^3 + 3x^2 + x - 5

Have fun doing some more!

Answer 4

x³ + x² + x + 1 = 0

Answer 5

If you are looking for an actuall equation, not a problem, I THINK this would work.

(X^3) + (15X^2) + (5x) + 7 ((numbers are interchangable))

^3 is cubed
^2 is squared. I don't know how to get the computer to do that.

Just make one of the cubed or squared one's without a number in front of the x, so that its just X.
THe other two can have numbers in front.

Like this ((written out)) 3Xcubed, + 5squared + 2X + 10.

Hope that is what your looking for.

Answer 6

(a +b)cube
=a cube +3a sq b +3a bsq +b cube

(a-b)cube
=a cube-3a sq b +3a bsq-bcube