Rational Root Theorem and Factor Theorem

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What is the Rational Root Theorem?

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The Rational Root Theorem states that if a polynomial has a rational root (p/q), then p (the numerator) is a factor of the constant term and q (the denominator) is a factor of the leading coefficient.

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How do you find possible rational zeros using the Rational Root Theorem?

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To find possible rational zeros, list all factors of the constant term (p) and all factors of the leading coefficient (q). The possible rational zeros are the ratios of these factors (p/q).

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What is the Factor Theorem?

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The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0.

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How can you use the Factor Theorem to find zeros of a polynomial?

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To find zeros, substitute potential zeros into the polynomial. If f(c) = 0, then (x - c) is a factor of the polynomial.

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List the possible rational zeros for h(x) = x^3 - 5x^2 + 2x + 12.

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±1, ±2, ±3, ±4, ±6, ±12.

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List the possible rational zeros for f(x) = 3x^5 - 5x^2 + x + 6.

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±1, ±2, ±3, ±6, ±1/3, ±2/3.

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List the possible rational zeros for p(x) = 3x^2 + x + 7.

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±1, ±7, ±1/3, ±7/3.

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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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How do you verify if a number is a zero of a polynomial?

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Substitute the number into the polynomial. If the result is zero, then it is a zero of the polynomial.

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What is the significance of finding rational zeros?

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Finding rational zeros helps in factoring polynomials, which can simplify solving polynomial equations.