Rational root theorem
Okay kiddo, do you know what a root is in math? It's basically the answer to an equation. For example, if we want to find the root of the equation x^2 - 4x + 3 = 0, we need to find a number that we can put in place of x that makes the equation true. In this case, the roots are 1 and 3, because if we put either of those numbers in for x, the equation will equal 0.
Now, the rational root theorem is a rule we can use to help us find the roots of certain equations. It tells us that if an equation has integer coefficients (that means all the numbers in the equation are whole numbers, like 2 or -5), then any rational root of the equation (that's a number that can be written as a fraction, like 1/2 or -3/4) must have a numerator that divides evenly into the constant term of the equation (that's the number by itself at the end of the equation) and a denominator that divides evenly into the leading coefficient of the equation (that's the number in front of the x^2 term).
Let's do an example to make it clearer. Say we have the equation 2x^3 - 5x^2 + 3x - 1 = 0. We want to find all the rational roots of this equation using the rational root theorem. First, we look at the constant term: -1. We need to find all the factors of -1, which are just 1 and -1 (since -1 times -1 = 1). These are the possible numerators of our rational roots.
Next, we look at the leading coefficient: 2. We need to find all the factors of 2, which are 1, 2, -1, and -2. These are the possible denominators of our rational roots.
So, all the possible rational roots of this equation are: 1/1, -1/1, 1/2, -1/2, 3/1, -3/1, 3/2, and -3/2. We can see which of these are actual roots of the equation by plugging them in and seeing if the equation equals 0. If we do this, we find that the only rational roots of the equation are 1/2 and -1/2.
And that's the rational root theorem in a nutshell, kiddo! It's just a way to help us find the rational roots of certain equations by looking at the factors of the constant term and leading coefficient.
Now, the rational root theorem is a rule we can use to help us find the roots of certain equations. It tells us that if an equation has integer coefficients (that means all the numbers in the equation are whole numbers, like 2 or -5), then any rational root of the equation (that's a number that can be written as a fraction, like 1/2 or -3/4) must have a numerator that divides evenly into the constant term of the equation (that's the number by itself at the end of the equation) and a denominator that divides evenly into the leading coefficient of the equation (that's the number in front of the x^2 term).
Let's do an example to make it clearer. Say we have the equation 2x^3 - 5x^2 + 3x - 1 = 0. We want to find all the rational roots of this equation using the rational root theorem. First, we look at the constant term: -1. We need to find all the factors of -1, which are just 1 and -1 (since -1 times -1 = 1). These are the possible numerators of our rational roots.
Next, we look at the leading coefficient: 2. We need to find all the factors of 2, which are 1, 2, -1, and -2. These are the possible denominators of our rational roots.
So, all the possible rational roots of this equation are: 1/1, -1/1, 1/2, -1/2, 3/1, -3/1, 3/2, and -3/2. We can see which of these are actual roots of the equation by plugging them in and seeing if the equation equals 0. If we do this, we find that the only rational roots of the equation are 1/2 and -1/2.
And that's the rational root theorem in a nutshell, kiddo! It's just a way to help us find the rational roots of certain equations by looking at the factors of the constant term and leading coefficient.
Related topics others have asked about: