A Polynomial can Have Zeros, but no Rational Zeros

f[x_]=x^2-2

Rational Zero Theorem:

Factors of -2: ±(1,2)
Factors of 1: ±1
Potential Rational Factors: ±(1,2)

Remainder Theorem to identify Factors:

f[1]
f[-1]
f[2]
f[-2]

So f(x) has no rational factors. But it does have real-valued zeros.

Plot[f[x],{x,-5,5},PlotStyle->Thick,BaseStyle->Large]
Solve[f[x]==0,x]

Factoring Higher Degree Polynomials

There is a quadratic formula:

Solve[a*(x^2)+b*x+c==0,x]

There is a cubic formula (that no one has memorized):

Solve[a*(x^3)+b*(x^2)+c*x+d==0,x]

There is even a quartic formula, but it is complicated:

Solve[a*(x^4)+b*(x^3)+c*(x^2)+d*x+e==0,x]

There is no quintic formula. We cannot factor quintic or higher polynomials in general (the fact no formula exists was proven in 1824).

Solve[a*(x^5)+b*(x^4)+c*(x^3)+d*(x^2)+e*x+f==0,x]

For this reason, it is important to have some process to allow us to find factors of polynomials in certain cases, which is what Sections 3.2 and 3.3 are all about.