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 possible rational roots using the Rational Root Theorem?

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To find 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) = 2x^3 - 11x^2 + 12x + 9?

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

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

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

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How can you determine if (x - 2) is a factor of f(x) = x^3 - 8x^2 + 14x - 4?

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You can use synthetic division or direct substitution. If f(2) = 0, then (x - 2) is a factor.

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What is synthetic division?

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

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What is the significance of the remainder in synthetic division?

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The remainder indicates whether the divisor is a factor of the polynomial. If the remainder is zero, the divisor is a factor.

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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 of the polynomial. 2. List all factors of the constant term (p) and leading coefficient (q). 3. Form all possible combinations of ±p/q.

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What is a complete list of possible rational zeros for f(x) = x^3 + 2x^2 - 6x + 8?

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The complete list of possible rational zeros is ±1, ±2, ±4, ±8.

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How many zeros are needed for synthetic division of x^6 + 4x^3 + 2?

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