Rational Root Theorem

Front

What is the Rational Root Theorem?

Back

The Rational Root Theorem states that any rational solution (or root) of a polynomial equation, in the form of p/q, is such that p is a factor of the constant term and q is a factor of the leading coefficient.

Front

What are the steps to apply the Rational Root Theorem?

Back

1. Identify the constant term and the leading coefficient of the polynomial. 2. List all factors of the constant term (p). 3. List all factors of the leading coefficient (q). 4. Form all possible rational roots as p/q.

Front

Given the polynomial f(x) = 2x^3 + 5x^2 - 9x + 5, what are the possible rational roots?

Back

±1, ±5, ±1/2, ±5/2.

Front

How do you determine the factors of a number?

Back

To determine the factors of a number, find all integers that can be multiplied together to produce that number.

Front

What is a rational root?

Back

A rational root is a solution to a polynomial equation that can be expressed as a fraction p/q, where p and q are integers and q is not zero.

Front

Solve the polynomial 2x^3 - 11x^2 + 12x + 9 = 0 using the Rational Root Theorem.

Back

The rational roots are -1/2 and 3.

Front

What is the significance of the leading coefficient in the Rational Root Theorem?

Back

The leading coefficient is used to determine the possible values of q in the rational roots p/q.

Front

What are the possible rational roots for the polynomial 3x^3 + 2x^2 - 6x + 7?

Back

±1, ±7, ±1/3, ±7/3.

Front

How can you verify if a number is a root of a polynomial?

Back

Substitute the number into the polynomial equation. If the result equals zero, then it is a root.

Front

What is the relationship between the Rational Root Theorem and synthetic division?

Back

Synthetic division can be used to test potential rational roots quickly and efficiently.