4.5: Factoring & Solving Polynomials Using the Rational Root Theorem

Goals for today:

Understand and be able to use the Rational Root Theorem to expedite factoring polynomials
Continue practice with Factoring (all sorts) & Solving

The Rational Root Theorem:

If a polynomial has integer coefficients, then every rational solution of $LaTeX: f\left(x\right)\:=\:0$ has the following form:

$LaTeX: \frac{p}{q}\:=\:\frac{factor\:of\:the\:constant}{factor\:of\:the\:leading\:coefficient}$

What does this mean?

If you have a polynomial $LaTeX: y\:=\:3x^5+\:7x^4-2x\:+8$

The possible integer factors could ONLY be 1, or 2, or 4, or 8. . . or they could be $LaTeX: \frac{1}{3},\:or\:\frac{2}{3},\:or\:\frac{4}{3},\:or\:\frac{8}{3}$

Why does this matter?

If you are trying to factor a polynomial, you don't have to test every number using synthetic division -- you can focus on only the numbers that might work! If none of the integer factors work, there are no rational roots for the polynomial.

Homework Practice: p. 194 #1-3; 5-23 odd; use Desmos or GC to graph.

Resources:

Factoring Sums & Differences of Cubes

Factoring by Grouping

Factoring Polynomials

Factoring & Solving Polynomials: Synthetic Division and the Remainder Theorem

Using the Rational Root Theorem to Find Zeros

Tutor.com: 24/7 -- link is on the left!

Remember you also have access to our Khan Academy class for notes & practice!