Rational root theorem

Learning Objectives State the Rational Root Theorem in its formal definition. Identify the constant term (a₀) and the leading coefficient (aₙ) of any polynomial. Generate a complete list of all possible rational roots for a given polynomial. Test potential rational roots efficiently using synthetic division and the Remainder Theorem. Use the Rational Root Theorem to find an initial root and then factor the polynomial completely. Solve polynomial equations of degree 3 or higher that have at least one rational root. Connect the rational roots of a polynomial equation P(x) = 0 to the x-intercepts of the graph of y = P(x). How can you find the exact x-intercepts of a complex cubic function like y = 2x³ - 5x² - 4x + 3 without a graphing calculator? 🤔 This tutorial introduces...

TermDefinitionExample Polynomial with Integer CoefficientsAn expression of the form P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where all the coefficients (aₙ, aₙ₋₁, ..., a₀) are integers and n is a non-negative integer.P(x) = 4x³ - 2x² + 5x - 10 is a polynomial with integer coefficients. Root (or Zero)A value 'c' for which a polynomial P(x) equals zero, i.e., P(c) = 0. Graphically, real roots are the x-intercepts of the polynomial function.For P(x) = x² - 4, the roots are x = 2 and x = -2 because P(2) = 0 and P(-2) = 0. Rational NumberAny number that can be expressed as a fraction p/q, where p and q are integers and q is not zero.3, -1/2, 0.75 (which is 3/4), and -5 (which is -5/1) are all rational numbers. Leading Coefficient (aₙ)The coefficient of the term with the highest power...

The Rational Root Theorem If the polynomial P(x) = aₙxⁿ + ... + a₀ has integer coefficients, then every rational root of P(x) = 0 can be written in the form p/q, where p is an integer factor of the constant term a₀, and q is an integer factor of the leading coefficient aₙ. Use this theorem to create a finite list of all possible rational roots for a polynomial equation. This is the starting point for finding the actual roots. The Factor Theorem A polynomial P(x) has a factor (x - c) if and only if P(c) = 0 (i.e., c is a root). This theorem connects finding a root to the process of factoring. Once you confirm that 'c' is a root using synthetic division (getting a remainder of 0), you know that (x - c) is a factor.