Prueba de raíces racionales

Prueba de raíces racionales

El Prueba de raíces racionales (also known as Rational Zeros Theorem) allows us to find all possible rational roots of a polynomial. Suppose a is root of the polynomial Pleft( x right) that means Pleft( a right) = 0. In other words, if we substitute a into the polynomial Pleft( x right) and get zero, 0, it means that the input value is a raíz de la función.

But how do we find the possible list of rational roots? Here‚Äôs how it works in a nutshell!



Key Ideas of Rational Roots Test

Suppose we have some polynomial Pleft( x right) with integer coefficients and a nonzero constant term:

Prueba de raíces racionales

Then every rational root of Pleft( x right) is of the form:


Prueba de raíces racionales

The best way to learn this method is to take a look at some examples!

Examples of How to Find the Rational Roots of a Polynomial using the Rational Roots Test

Ejemplo 1: Find the rational roots of the polynomial below using the Rational Roots Test.


Prueba de raíces racionales

Finding the rational roots (also known as rational zeroes) of a polynomial is the same as finding the rational x-intercepts.


  • Start by identifying the constant term a0 and the leading coefficient an.
Prueba de raíces racionales
  • Determine the positive and negative factors of each.

Factors of constant term, {a_0} = 6,,:,, pm ,left( {1,2,3,6} right)

Factors of leading term, {a_n} = 3,,:,, pm ,left( {1,3} right)

  • Write down the list of the possible rational roots by finding {p over q} which is simply the proporci√≥n of the factors of the t√©rmino constante y t√©rmino principal. Make sure that you keep track of the possible combinations.

This is how I do it. I take each numerator and divide it by all denominators. Then I move on to the next numerator and again divide by all denominators. I keep repeating this process until I have gone through all the numerators. This ensures that we have covered all possible combinations.



Prueba de raíces racionales

BIG Caution: After you write down all combinations, simplificar the fractions in order to get rid of duplicates.

Prueba de raíces racionales

So these are the numbers without duplicates that we will check as possible roots. We have twelve (12) possible candidates to check.

Prueba de raíces racionales
  • Remember that if a is a root of the polynomial Pleft( x right), then Pleft( a right) = 0. Now, let‚Äôs check each number.
Prueba de raíces racionales
  • Therefore, the rational roots of the polynomial
Prueba de raíces racionales

Médica


Prueba de raíces racionales

Here is the graph of the polynomial showing where it crosses or touches the x-axis. These are in fact the x-intercepts of the polynomial.

Prueba de raíces racionales

Ejemplo 2: Find the rational roots of the polynomial below using Rational Roots Test.

Prueba de raíces racionales

The constant term is a0 = ‚Äď2 and its possible factors are p = ¬Ī 1, ¬Ī 2. For the leading coefficient, we have an = 4 and its factors are q = ¬Ī 1, ¬Ī 2, ¬Ī 4.

  • To find the possible roots of the polynomial, write in the form
Prueba de raíces racionales Prueba de raíces racionales

Write down all possible 

Prueba de raíces racionales

combinaciones:

Prueba de raíces racionales
  • Simplify each fraction to eliminate duplicates or identical values. Here‚Äôs our new and improved list!
Prueba de raíces racionales
  • Due to the plus or minus consideration of each number, we will have eight (8) possible candidates as the roots of this polynomial.
Prueba de raíces racionales

If you plug in each value to the given polynomial and gets zero, that means the number you substituted is a root! Try this on paper, and you should be convinced that there are only three values satisfying this condition.

Prueba de raíces racionales

Therefore, the rational roots of the polynomial

Prueba de raíces racionales

 M√©dica

Prueba de raíces racionales

Graphically, it shows that the polynomial touches or crosses the x-axis at those roots determined by rational roots test.

Prueba de raíces racionales

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