Rational root theorem facts for kids

The Rational Root Theorem is a cool idea in mathematics that helps you find possible solutions for certain types of math problems called polynomial equations. It's also sometimes called the Rational Zero Theorem.

Imagine you have a polynomial equation, which is a math expression with different powers of a variable (like x, x², x³, etc.). If all the numbers in front of these variables (called coefficients) and the number at the very end (called the constant term) are whole numbers (like 1, 2, -5, 0), then this theorem can help you find any solutions that are rational numbers. Rational numbers are numbers that can be written as a fraction, like 1/2 or 3/4.

The theorem tells you that any rational solution must be a fraction made by dividing a factor of the constant term by a factor of the leading coefficient. The leading coefficient is the number in front of the variable with the highest power.

What is a Polynomial?

A polynomial is a math expression that looks like this: a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀

Let's break that down:

• x is the variable, usually a letter like x or y.
• n is the highest power of x in the equation. It must be a whole number (0, 1, 2, 3, ...).
• a_n, a_n-1, ... a₀ are the numbers in front of the x terms. These are called coefficients.
• a_n is the leading coefficient. It's the number in front of the x with the highest power.
• a₀ is the constant term. It's the number without any x next to it.

For the Rational Root Theorem to work, all these coefficients (a_n, a_n-1, etc.) and the constant term (a₀) must be integers. Integers are whole numbers, including positive numbers, negative numbers, and zero (like -3, -2, -1, 0, 1, 2, 3).

What are Rational Roots?

When we talk about "roots" of a polynomial, we mean the values of x that make the entire polynomial equal to zero. For example, if you have the polynomial x - 2, the root is 2 because 2 - 2 = 0.

A rational root is a root that can be written as a fraction (a/b), where 'a' and 'b' are integers, and 'b' is not zero. For example, 1/2, -3/4, or 5 (which can be written as 5/1) are all rational numbers.

The Rational Root Theorem helps you find a list of possible rational roots. It doesn't guarantee that any of them are actual roots, but it narrows down the search a lot! You still need to test these possible roots to see if they work.

How Does the Theorem Work?

The theorem states that if a polynomial has integer coefficients, then any rational root (let's call it p/q) must follow these rules:

• p must be a factor of the constant term (a₀).
• q must be a factor of the leading coefficient (a_n).

Remember that factors are numbers that divide evenly into another number. For example, the factors of 6 are 1, 2, 3, and 6 (and their negative versions: -1, -2, -3, -6).

So, to find all possible rational roots, you just need to:

1. Find all the factors of the constant term (a₀).
2. Find all the factors of the leading coefficient (a_n).
3. Create all possible fractions by putting a factor from step 1 over a factor from step 2. Don't forget to include both positive and negative versions (±).

Step-by-Step Example

Let's look at an example to see how this works. Consider this polynomial: 15x⁴ + 2x³ – 10x² + x – 8

Here's how we find the possible rational roots:

1. Identify the leading coefficient and the constant term:

* The leading coefficient (a_n) is 15 (the number in front of x⁴). * The constant term (a₀) is -8 (the number at the end).

1. Find the factors of the leading coefficient (15):

* The factors of 15 are: ±1, ±3, ±5, ±15.

1. Find the factors of the constant term (-8):

* The factors of -8 are: ±1, ±2, ±4, ±8.

1. List all possible rational roots (p/q):

Now, we make all possible fractions by dividing each factor of -8 by each factor of 15. * Dividing by ±1: ±1/1, ±2/1, ±4/1, ±8/1 * Dividing by ±3: ±1/3, ±2/3, ±4/3, ±8/3 * Dividing by ±5: ±1/5, ±2/5, ±4/5, ±8/5 * Dividing by ±15: ±1/15, ±2/15, ±4/15, ±8/15

So, the list of all possible rational roots is: ±1, ±2, ±4, ±8, ±1/3, ±2/3, ±4/3, ±8/3, ±1/5, ±2/5, ±4/5, ±8/5, ±1/15, ±2/15, ±4/15, ±8/15.

This theorem only helps you find the rational roots. Polynomials can also have imaginary roots (roots that involve the square root of negative numbers). Other math tools, like Descartes' Rule of Signs, can help you figure out how many imaginary roots an equation might have.

Why is This Theorem Useful?

The Rational Root Theorem is very useful because it gives you a starting point when you're trying to solve polynomial equations. Without it, finding the roots could be like looking for a needle in a haystack. This theorem helps you narrow down the possibilities to a manageable list of numbers to test. Once you find one rational root, you can often use other methods (like polynomial long division) to simplify the polynomial and find the remaining roots.

See also

• In Spanish: Teorema de la raíz racional para niños