Expressions relating to roots of polynomials

Polynomial equations have roots that satisfy the equation. For example, if x=a is the root of the polynomial equation P(x) = 0, substituting x=a would make P(a)=0.
For a quadratic equation ax^2 + bx + c = 0, the sum of the roots can be determined by the ratio -b/a and the product of the roots can be determined by the ratio c/a.
This extends to cubic and higher degree polynomials. For a cubic equation ax^3 + bx^2 + cx + d = 0, the sum of roots is -b/a, the sum of the products of roots taken two at a time is c/a, and the product of all roots is -d/a.
Using the coefficients of a polynomial equation and the relationships mentioned above, we can form equations to find unknown roots.
The ‘Factor Theorem’ and ‘Remainder Theorem’ are useful for manipulating polynomial expressions and finding roots. The Factor Theorem states that if P(a)=0 for certain polynomial P(x), then (x-a) must be a factor. The Remainder Theorem asserts that when a polynomial P(x) is divided by (x-a), the remainder is equal to P(a).
Complex numbers may be roots of polynomials. For polynomials with real coefficients, if there is a complex root, its conjugate will also be a root.
We can derive an expression that relates the roots of the derivative of a polynomial to the roots of the polynomial itself. Remember, if a root of a polynomial is also a root of its derivative, that root is known as a ‘repeated’ root.
If a polynomial equation has no real roots, this implies that the graph of the polynomial does not intersect the x-axis.
The ‘Fundamental Theorem of Algebra’ insists that every n-degree polynomial has exactly n roots, counting repetitions and including both real and complex roots.
The ‘Rational Root Theorem’ can be useful in finding roots of polynomials. It states that if a polynomial has integer coefficients and a rational root p/q (where p and q are coprime), p must be a factor of the constant term of the polynomial and q must be a factor of the leading coefficient.