Theorems for Finding Roots of Polynomials
Roots are the \(x\)-intercepts of the graph of \(f(x)\). Make a note of how many \(x\)-intercepts were in each of the new graphs you created. These \(x\)-intercepts are the polynomial’s real roots, but not all roots are real. Some are imaginary, so you won’t be able to see them in a graph on the \(xy\) plane.
As you remember, the Fundamental Theorem of Algebra states:
Every polynomial equation of degree \(n\) with complex coefficients has \(n\) roots in the complex numbers.
You previously learned how to factor a quadratic, what if the polynomial has a greater degree than two?
Some tools that are very useful in determining the roots for polynomials with degrees greater than two are:
1) The Remainder Theorem
2) The Rational Roots Theorem
3) The Factor Theorem
4) Synthetic Division
Synthetic division is not new to you, but the Rational Roots Theorem and The Factor Theorem may be. The definitions are below.
The Rational Roots Theorem states: If \(P(x)\) is a polynomial with integer coefficients and if \(\dfrac{p}{q}\) is a zero of \(P(x)\) \((P\bigg(\dfrac{p}{q}\bigg)=0)\), then \(p\) is a factor of the constant term of \(P(x)\) and \(q\) is a factor of the leading coefficient of \(P(x)\). This allows us to find potential roots of a polynomial without starting for scratch. Although, the rational roots theorem may yield far more potential roots than actually work.
The Factor Theorem: When \(f(c)=0\) then \(x-c\) is a factor of the polynomial And the other way around, too: When \(x-c\) is a factor of the polynomial then \(f(c)=0\). Watch the video linked below and then return to this page by using the back arrow.
Using the Polynomial Remainder Theorem: checking factors (video) | Khan Academy
Remainder Theorem: If the polynomial \(f(x)\) is divided by \(x-c\), then the remainder is \(f(c)\). Watch the video linked below and then return to this page by using the back arrow.
Intro to the Polynomial Remainder Theorem (video) | Khan Academy
