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Understanding Rational Roots, Remainder, and Factor Theorems in Polynomials

This guide explores three fundamental theorems in polynomial algebra: the Rational Roots Theorem, the Remainder Theorem, and the Factor Theorem. The Rational Roots Theorem states that all possible rational roots of a polynomial can be expressed as p/q, where p are the factors of the constant term and q are the factors of the leading coefficient. The Remainder Theorem indicates that dividing a polynomial P(x) by x - r leaves a constant remainder of P(r). Finally, the Factor Theorem asserts that (x - r) is a factor of P(x) if and only if P(r) = 0.

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Understanding Rational Roots, Remainder, and Factor Theorems in Polynomials

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  1. Rational Roots Theorem: All possible rational roots of a polynomial are of the form p/q, where p = the factors of the constant term and q = the factors of the leading coefficient. Remainder Theorem: If a polynomial P(x) is divided by x – r, then the remainder is a constant = P(r). Factor Theorem: The binomial (x – r) is a factor of the polynomial P(x) if and only if P(r) = 0. Theorems for 4.3

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