# What does the rational zero theorem say?

## What does the rational zero theorem say?

The Rational Zeros Theorem states: If P(x) is a polynomial with integer coefficients and if is a zero of P(x) (P( ) = 0), then p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x).

### What is considered a rational zero?

A rational zero is a rational number, which is a number that can be written as a fraction of two integers. An irrational zero is a number that is not rational, so it has an infinitely non-repeating decimal.

#### Why is it helpful to use the Rational Zero Theorem in finding the zeros of a polynomial function?

It is used to find out if a polynomial has rational zeros/roots. It also gives a complete list of possible rational roots of the polynomial. It also comes in handy when we need to factor a polynomial alongside with the use of polynomial long division or synthetic division.

How do you find possible zeros?

Use the Rational Zero Theorem to list all possible rational zeros of the function. Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. If the remainder is 0, the candidate is a zero. If the remainder is not zero, discard the candidate.

What is the difference between rational and real zeros?

Explanation: Rational are those numbers which can be written as a ratio of two integers, the denominator being non-zero. Real numbers are those, which can be represented on real number line.

## Can a real zero be negative?

Note how there are no sign changes between successive terms. This means there are no negative real zeros. Since we are counting the number of possible real zeros, 0 is the lowest number that we can have.

### How many real zeros are there?

Assuming the polynomial is non-constant and has Real coefficients, it can have up to n Real zeros. If n is odd then it will have at least one Real zero. Since any non-Real Complex zeros will occur in Complex conjugate pairs the possible number of Real roots counting multiplicity is an even number less than n .

#### Which is correct zeros or zeroes?

According to the Merriam-Webster Dictionary, the plural of “zero” can either be “zeros” or “zeroes.” They are both correct.

What is the rational Zero Theorem?

The Rational Zero Theorem. The Rational Zero Theorem gives a list of possible rational zeros of a polynomial function. Equivalently, the theorem gives all possible rational roots of a polynomial equation.

How do you find the rational zeros of a function?

Rational Zero Theorem. If a polynomial function, written in descending order of the exponents, has integer coefficients, then any rational zero must be of the form ± p / q, where p is a factor of the constant term and q is a factor of the leading coefficient. Example 1. Find all the rational zeros of. f ( x) = 2 x 3 + 3 x 2 – 8 x + 3.

## How do you factor polynomials using the rational zeros theorem?

We can often use the rational zeros theorem to factor a polynomial. Using synthetic division, we can find one real root a and we can find the quotient when P(x) is divided by x – a.

### What are the requirements for a rational zero?

According to the rational zero theorem, any rational zero must have a factor of 3 in the numerator and a factor of 2 in the denominator. These values can be tested by using direct substitution or by using synthetic division and finding the remainder.