Can you please explain the rational root theorem and the factor theorem?

Question

Genesis Allen · Accepted Answer

Both theorems are discussed in Precalculus.

Rational Root Theorem

Factor Theorem

(It would make sense for them to be discussed in Algebra also, but they aren't on the topic list. At least, I didn't find them.)

Ethan Adkins · Answer

undefined The Rational Root Theorem states that if a polynomial equation has rational roots (solutions), then those roots must be among the possible factors of the constant term divided by the possible factors of the leading coefficient. For example, if we have a polynomial equation $ ax^n + bx^{n-1} + \ldots + cx + d = 0 $, where $ a $, $ b $, $ c $, and $ d $ are integers, then any rational roots of this equation must be of the form $ \frac{p}{q} $, where $ p $ is a factor of the constant term $ d $ and $ q $ is a factor of the leading coefficient $ a $.

The Factor Theorem is an extension of the Rational Root Theorem and states that if a polynomial $ f(x) $ has a root $ r $, then $ (x - r) $ is a factor of $ f(x) $. In other words, if $ f(r) = 0 $, then $ (x - r) $ is a factor of $ f(x) $. This theorem provides a useful method for factoring polynomials and finding their roots. By using synthetic division or polynomial long division, we can divide a polynomial by $ (x - r) $ to find the other factors and ultimately factorize the polynomial completely.

In summary, the Rational Root Theorem helps identify potential rational roots of a polynomial equation, while the Factor Theorem provides a method for factoring polynomials by identifying factors corresponding to their roots. Both theorems are fundamental tools in algebra for solving polynomial equations and understanding their properties.