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The Rational Root Theorem: Explanation and Proof
Rene Descartes has been credited with discovering the Rational Root Theorem. Source.
A brief explanation and proof
The Rational Root Theorem (RRT) is a handy tool to have in your mathematical arsenal. It provides and quick and dirty test for the rationality of some expressions. And it helps to find rational roots of polynomials.
Here’s how and why it works.
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Suppose you have a polynomial of degree n, with integer coefficients:
The Rational Root Theorem states: If a rational root exists, then its components will divide the first and last coefficients:
The rational root is expressed in lowest terms. That means p and q share no common factors. (That will be important later.) The numerator divides the constant at the end of the polynomial; the demominator divides the leading coefficient.
As an example:
We need only look at the 2 and the 12. If a rational root p/q exists, then:
The factors of 12 are:
The factors of 2:
Thus, if a rational root does exist, it’s one of these:
Plug each of these into the polynomial. Which one(s) — if any solve the equation? If none do, there are no rational roots.
Are any cube roots of 2 rational? A rational root, p/q must satisfy this equation.
Not one of these candidates qualifies. So:
Let’s go back to our paradigm polynomial.
Scoot the constant to the other side:
Now, plug in our rational root, p/q.
Multiply everything by qⁿ:
Each term on the left has p in common. Factor that out.
It looks a lot worse than it needs to be. Let’s replace all that stuff in parenthesis with an s. We don’t really care what’s in there.
That’s much easier on the eyes.
Remember that p and q are integers. They also share no common factors. Therefore, p cannot divide qⁿ. It must divide a₀:
Thus, the numerator divides the constant term.
Now, go back to our paradigm polynomial:
This time, move the first term to the right side.
Insert the rational root:
As before, multiply by qⁿ.
This time, the common factor on the left is q. Let’s extract it, and lump together the remaining sum as t.
Again, q and p have no common factors. Therefore:
Thus proves the rational root theorem.
5.4.4.docx – According to the rational root theorem which – According to the rational root theorem, which of the following are possible roots of the polynomial function below? F (x) = 6 x 3 – 7 x 2 + 2 x + 8 A. 3 B. 4 C.-8 D. E. F.-At most, how many unique roots will a fourth-degree polynomial function have? A. 8 B. 4 C. 3Rational Roots Test. The Rational Roots Test (also known as Rational Zeros Theorem) allows us to find all possible rational roots of a polynomial. Suppose a is root of the polynomial P\left( x \right) that means P\left( a \right) = 0.In other words, if we substitute a into the polynomial P\left( x \right) and get zero, 0, it means that the input value is a root of the function.Rational Root Theorem. The Rational Root Theorem from Algebra is used to find possible zeros of a polynomial function of order 3. The Rational Root Theorem is an important tool used to investigate
Rational Roots Test – ChiliMath – The rational root theorem describes a relationship between the roots of a polynomial and its coefficients. Specifically, it describes the nature of any rational roots the polynomial might possess. Let's work through some examples followed by problems to try yourself. Submit your answer A polynomial with integer coefficientsQuestion: According to the Rational Root Theorem, -2/5 is a potential rational root of which function? f(x) = 4×4 – 7×2 + x + 25 f(x) = 9×4 – 7×2 + x + 10 f(x) = 10×4 – 7×2 + x + 9 f(x) = 25×4 – 7×2 + x + 4Find an answer to your question According to the Rational Root Theorem, -7/8 is a potential rational root of which function? A) f(x) = 24×7 + 3×6 + 4×3 – x – 28…
According to the Rational Root Theorem, which of the – Rational root theorem, also called rational root test, in algebra, theorem that for a polynomial equation in one variable with integer coefficients to have a solution that is a rational number, the leading coefficient (the coefficient of the highest power) must be divisible by the denominator of the fraction and the constant term (the one without a variable) must be divisible by the numerator.Question: According to the Rational Root Theorem, the following are potential roots of f(x) = 2×2 + 2x – 24. -4, -3, 2, 3, 4 Which are actual roots of f(x)? A) -4 and 3 B) -4, 2, and 3 C) -3 and 4 D) -3, 2, and 4This video shows how to find the rational roots of a polynomial by the rational root theorem and synthetic division. Two examples are shown.