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, must have p as a factor of the constant term and q as a factor of the leading coefficient.

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How do you find the possible rational zeros of a polynomial?

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To find the possible rational zeros of a polynomial, list all factors of the constant term (p) and all factors of the leading coefficient (q). The possible rational zeros are then the combinations of these factors in the form ±p/q.

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What are the possible rational zeros of f(x) = 2x^3 + 5x^2 - 9x + 5?

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±1, ±5, ±1/2, ±5/2.

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Is (x-2) a factor of f(x) = x^3 - 8x^2 + 14x - 4?

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Yes, (x-2) is a factor. The remainder is zero.

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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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How do you determine if a number is a root of a polynomial?

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To determine if a number is a root of a polynomial, substitute the number into the polynomial. If the result is zero, then the number is a root.

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What is the significance of the leading coefficient in the Rational Root Theorem?

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The leading coefficient is important because it helps determine the possible values of q in the rational root p/q.

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Solve for x: x^3 + 3x^2 - 6x - 8 = 0 using the Rational Root Theorem.

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The solutions are x = -4, -1, 2.

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What does it mean if a polynomial has a remainder of zero when divided by (x-c)?

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If a polynomial has a remainder of zero when divided by (x-c), it means that (x-c) is a factor of the polynomial and c is a root.

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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.