Roots and factors of a polynomial
Let P(x) be a polynomial. A number a such that P(a)=0 is called a root or zero of P.
If x=a is a root of a polynomial, then x-a is a factor of that polynomial.
Consider the polynomial P(x)=x2+2x-3.
Let's plug x=1 into the polynomial: p(1)=12+2·1-3=0.
Consequently x=1 is a root of the polynomial P(x)=x2+2x-3.
Note that x-1 is a factor of the polynomial P(x)=x2+2x-3.
The fundamental theorem of algebra stated that a polynomial P(x) of degree n has n roots, some of which may be degenerate.
What are the possible integer roots of P(x)=2x5 -3x3 +4x2 -9x + 6 ?
If there are integer roots, they will be factors of the constant term 6; namely 1, -1, 2, -1, 3, -2, 6, -6.
Now, is 1 a root? To answer, we will divide the polynomial by x-1 and hope for remainder 0. Yes! 1 is a root!
Now, is 1 a root again? To answer, we divide the resulting polynomial by x-1 and hope for remainder 0. Yes! 1 is a root again!.
Now, is -2 a root? Yes! -2 is a root!
We have: 2x5 -3x3 +4x2 -9x + 6 =(x-1)2(x+2)(2x2 +3)where 2x2 +3 does not have integer roots.
Conclude that the integer roots of P(x)=2x5 -3x3 +4x2 -9x + 6 are a=1, b=1 and c=-2