Rational Root Theorem

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What is the Rational Root Theorem?

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The Rational Root Theorem states that any rational solution (or root) of a polynomial equation, in the form of p/q, is such that p is a factor of the constant term and q is a factor of the leading coefficient.

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How do you determine possible rational roots using the Rational Root Theorem?

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List all factors of the constant term (p) and all factors of the leading coefficient (q). The possible rational roots are then the combinations of these factors in the form p/q.

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What is a factor of a polynomial?

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A factor of a polynomial is a polynomial of lower degree that divides the polynomial without leaving a remainder.

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What does it mean if (x - r) is a factor of f(x)?

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If (x - r) is a factor of f(x), then f(r) = 0, meaning r is a root of the polynomial.

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What is synthetic division?

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Synthetic division is a simplified form of polynomial long division that is used to divide a polynomial by a linear factor of the form (x - r).

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How do you set up synthetic division?

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Write the coefficients of the polynomial in a row. If there are missing degrees, use 0 as the coefficient for those degrees. Use the root r to the left of the coefficients.

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What is the remainder theorem?

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The remainder theorem states that the remainder of the division of a polynomial f(x) by (x - r) is equal to f(r).

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What is the significance of the remainder being zero in polynomial division?

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If the remainder is zero, it indicates that (x - r) is a factor of the polynomial f(x).

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What are the steps to apply the Rational Root Theorem?

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1. Identify the constant term and leading coefficient. 2. List all factors of the constant term (p) and leading coefficient (q). 3. Form possible rational roots as p/q. 4. Test these roots in the polynomial.

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What is a complete list of possible rational zeros?

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A complete list of possible rational zeros includes all combinations of the factors of the constant term and the leading coefficient, expressed as p/q.