Rational Root Theorem & Finding Roots Cheat Sheet

Key Facts

• For a polynomial $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ with integer coefficients, every rational root has the form $\frac{p}{q}$ where $p \mid a_0$ and $q \mid a_n$.
• The possible rational roots are $\pm \frac{\text{factors of the constant term}}{\text{factors of the leading coefficient}}$ after reducing duplicates.
• If $f(r) = 0$, then $r$ is a root, $x = r$ is a zero, and $x - r$ is a factor of $f(x)$.
• Synthetic division by $r$ gives a remainder equal to $f(r)$, so a remainder of $0$ means $r$ is a root.
• If $x - r$ is a factor of $f(x)$, then $f(x) = (x - r)q(x)$ for some quotient polynomial $q(x)$.
• A polynomial of degree $n$ has at most $n$ real roots and exactly $n$ complex roots when multiplicity is counted.
• A repeated root such as $x = r$ from the factor $(x - r)^2$ has multiplicity $2$ and may touch the $x$-axis instead of crossing it.
• After using the Rational Root Theorem, remaining quadratic factors can often be solved by factoring, completing the square, or $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Vocabulary

Rational Root

A rational root is a solution to a polynomial equation that can be written as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

Leading Coefficient

The leading coefficient is the coefficient of the highest-degree term in a polynomial, such as $a_n$ in $a_nx^n + \cdots + a_0$.

Constant Term

The constant term is the term with no variable, such as $a_0$ in $a_nx^n + \cdots + a_0$.

Zero

A zero of a function is an input value $r$ that makes the output equal to zero, so $f(r) = 0$.

Factor Theorem

The Factor Theorem states that $x - r$ is a factor of $f(x)$ if and only if $f(r) = 0$.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear factor $x - r$ and checking the remainder.