Finding Roots of Polynomial Functions Using ±(p/q) and Synthetic Division

We will walk through a process for solving for the roots (and factors) of the polynomial f(x) = x^4 -3x^3 + 5x^2 - x -10 using possible rational roots (± p/q) and synthetic division. This polynomial has two real roots and two roots that are complex conjugates.

[@Explicitzz_]

Thank you so much Dr. Pierce.

[@helloitsdp]

Thank you so much for this. We payed my summer school teacher 3k dollars to help me with algebra 2. She teaches us very hard methods and you just solved this problem for me in the easiest possible way. Thank you so much.

[@helloitsdp]

Quick question can we use the first method of finding possible roots and plug each one into the equation to find all the roots or does it only work once?

[@dr.piercesphysicsmath9071]

It can find more than one rational root if the polynomial has more than one rational root. It will not find irrational or complex roots.
Another strategy: if you find a rational root r, then (x-r) is a factor. You can divide the original polynomial by (x-r) and then work on finding the roots of the quotient, which will be one degree smaller (easier to solve!) than the original polynomial.

[@cosmiceggs2385]

When plugging in, it has to equal 0 to be considered a root?

[@dr.piercesphysicsmath9071]

Yep! A root of, say. x = 2 means both that the function y(x) is zero when x = 2 and that (x-2) is a factor with a remainder of zero.

[@cosmiceggs2385]

@@dr.piercesphysicsmath9071 thank you!