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 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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What is the Factor Theorem?

Back

The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0, meaning c is a root of the polynomial.

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

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List all factors of the constant term (p) and the leading coefficient (q). The possible rational roots are the ratios of these factors (p/q).

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Given f(x) = x^3 + 2x^2 - 11x - 12, find one rational root.

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The rational root is -4.

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List all possible rational roots for f(x) = x^3 + 2x^2 - 6x + 8.

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±1, ±2, ±4, ±8.

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

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±1, ±3, ±9, ±1/2, ±3/2, ±9/2.

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

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-1/3, -7/3, -1, -7, 1/3, 7/3, 1, 7.

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

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

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What is a polynomial?

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A polynomial is a mathematical expression consisting of variables, coefficients, and exponents, combined using addition, subtraction, multiplication, and non-negative integer exponents.

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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.