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 roots of a polynomial?

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To find the possible rational roots, 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 are the possible rational roots of the polynomial f(x) = 3x^3 + 2x^2 - 6x + 7?

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The possible rational roots are ±1, ±7, ±1/3, ±7/3.

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What are the possible rational roots of the polynomial f(x) = x^3 + 2x^2 - 6x + 8?

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The possible rational roots are ±1, ±2, ±4, ±8.

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How can you determine if (x - c) is a factor of a polynomial f(x)?

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You can determine if (x - c) is a factor of f(x) by using synthetic division or by evaluating f(c). If f(c) = 0, then (x - c) is a factor.

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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 are the possible rational roots of the polynomial 2x^3 - 11x^2 + 12x + 9 = 0?

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The possible rational roots are ±1, ±3, ±9, ±1/2, ±3/2, ±9/2.

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When performing synthetic division, how many zeros do you need for the polynomial x^6 + 4x^3 + 2?

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You need 4 zeros when setting up the top row of your synthetic division.

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What is a rational zero?

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A rational zero is a solution to a polynomial equation that can be expressed as a fraction p/q, where p and q are integers.

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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 - c).