The Rational Zero Theorem | Finding Possible Rational Zeros of a Polynomial Function

Rational Zero Theorem

The Rational Zero Theorem is a theorem in algebra that helps us find possible rational zeros of a polynomial function

The Rational Zero Theorem is a theorem in algebra that helps us find possible rational zeros of a polynomial function. A polynomial function is an equation that contains one or more terms, where each term is the product of a constant coefficient and one or more variables raised to non-negative integer exponents. The Rational Zero Theorem is useful when we want to find the rational solutions (zeros) of a polynomial equation.

The theorem states that if a polynomial function has a rational zero, then that zero must be of the form p/q, where p is an integer factor of the constant term and q is an integer factor of the leading coefficient. In other words, the Rational Zero Theorem gives us a list of possible rational zeros for a polynomial equation to check.

Let’s say we have a polynomial function f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where aₙ, aₙ₋₁, …, a₁, a₀ are the coefficients, with aₙ ≠ 0. According to the Rational Zero Theorem, the possible rational zeros for this function are ±(factor of a₀) / (factor of aₙ).

To find the rational zeros, we can use synthetic division or long division to test each potential zero (p/q) from the list. If a particular value p/q results in a remainder of zero, then it is a rational solution or zero of the polynomial equation.

It is important to note that the Rational Zero Theorem only provides possible rational zeros, not necessarily all the zeros of a polynomial function. There may be additional irrational or complex zeros that are not given by this theorem.

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