Rational Root Theorem 2024

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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 apply the Rational Root Theorem to find possible rational roots?

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To apply the Rational Root Theorem, list all factors of the constant term (p) and 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 for 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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Solve the polynomial x^3 + 3x^2 - 6x - 8 = 0 using the Rational Root Theorem.

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The roots are -4, 2, 1.

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What is a zero of a polynomial?

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A zero of a polynomial is a value of x that makes the polynomial equal to zero.

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How can you verify if a number is a root of a polynomial?

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To verify if a number is a root, substitute it into the polynomial equation. If the result is zero, then it is a root.

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What is the significance of finding rational roots in polynomial equations?

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Finding rational roots helps in factoring the polynomial, simplifying the equation, and solving for other roots.

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What are the steps to use 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 the 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 do you find the actual roots of a polynomial after identifying possible rational roots?

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Test each possible rational root in the polynomial. If it yields zero, it is an actual root. Use synthetic division to factor the polynomial further.