Question

What are the possible number of positive real, negative real, and complex zeros of `f(x) = 4x^3 + x^2 + 10x – 14`?

Answer 1

This cubic has one Real zero and a Complex conjugate pair of non-Real zeros.

Explanation:

`f(x) = 4x^3+x^2+10x-14`

The signs of the coefficients follow the pattern: `+ + + -`

With one change of sign, we can tell that this cubic polynomial has one positive zero.

Consider `f(-x)`...

`f(-x) = -4x^3+x^2-10x-14`

The signs of the coefficients follow the pattern: `- + - -`

With two changes of sign, we can tell that this cubic has `0` or `2` negative zeros.

Let us examine the discriminant:

`4x^3+x^2+10x-14` is in the form `ax^3+bx^2+cx+d` with `a=4`, `b=1`, `c=10` and `d=-14`.

This has discriminant `Delta` given by the formula:

`Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd`

`=100-16000+56-84672-10080`

`=-110596`

Since `Delta < 0` we can deduce that `f(x)` has one Real zero and a pair of Complex conjugate zeros.

graph{4x^3+x^2+10x-14 [-10, 10, -200, 200]}