Understanding How To Find Zeroes Of Polynomials.

Question

I am currently learning how to find zeroes of polynomials. However, 4 things confuse me.

1. The first question I have is why exactly does the rational zeroes theorem hold true? I get how to use it, but I don't get why it is true. Can someone please explain this?

2. The second question I have is why is the Descartes rule of signs true?

3. My third question is how do we find the imaginary roots of polynomial? It can't be through the rational zeroes theorem! But if we don't need to find imaginary roots, why else would we need things like the conjugate pairs theorem?

4. My last question is of synthetic division in finding the zeroes of polynomials. I don't get why using synthetic division to test if a root is a zero of a polynomial is by taking it, and synthetically dividing it. What I don't get is, why don't you need to change the sign of the zero before synthetically dividing here? Let's say I suspected that the zero of a polynomial is x=-4. Don't I need to change the sign of the root to x+4=0, then synthetically divide by 4? Why is that you don't need to do this?

Can you explain all of this using simple algebra, without complicated techniques? I don't understand any complicated techniques and theorems beyond the quadratic formula. Can you also show and explain your working, so it is easier for me to follow through? I am still a beginner, so that would help very much.

Answer 1

1. The rational roots or rational zeros theorem only applies to polynomials that have integer coefficients - although a polynomial with rational coefficients can be turned into a polynomial with integer coefficients by multiplying through by the LCM of the coefficients' denominators. The simplest way to see why it is true is to notice that if $\frac{p}{q}$ is a root of the polynomial then $qx-p$ is a factor of the polynomial and so $q$ must be a factor of the coefficient of the highest power term and $p$ must be a factor of the constant term.
2. There is a simple proof of Descartes rule of signs here: https://www.math.hmc.edu/funfacts/ffiles/20001.1.shtml. It uses induction of the degree of the polynomial, and compares the sign changes in a polynomial with the sign changes in its derivative.
3. There are explicit formulae for finding the roots of cubic and quartic polynomials (similar to the quadratic formula but more complicated). Finding complex roots of higher degree polynomials is difficult, and usually requires an iterative algorithm such as Newton's method to approximate the roots.
4. I'm not sure I understand this question. To test whether $-4$ is a root of a polynomial you can just substitiute $x=-4$ to see whether $f(-4)=0$. If you know $-4$ is a root of the polynomial then you can find the quotient after you have removed that root by dividing the polynomial by $x+4$. You know $x+4$ know must be a factor of the polynomial because $f(-4)=0$.

Answer 2

The reason in the fourth question is that we can always divide one polynomial by another leaving a remainder of lower degree, so that dividing by a linear polynomial gives a constant remainder. Suppose we divide by $(x-a)$ so that $$p(x)=(x-a)q(x)+r$$

If we then set $x=+a$ we get $$p(a)=r$$ and if $a$ is a root we have $r=0$.

The signs work as they do because we need to make the $(x-a)q(x)$ factor disappear - we are not interested in $q(x)$ - and we can make this happen by setting $x-a=0$