Rational Root Theorem and Factor Theorem

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What does the Rational Root Theorem state?

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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, is such that p is a factor of the constant term and q is a factor of the leading coefficient.

Front

How do you find all possible rational roots of a polynomial?

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To find all possible rational roots of a polynomial, identify the factors of the constant term (p) and the leading coefficient (q), then form all possible combinations of p/q.

Front

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

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±1, ±3, ±9, ±1/2, ±3/2, ±9/2.

Front

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

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1/3, 2/3, 1, 2, 3, 6, -1/3, -2/3, -1, -2, -3, -6.

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

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

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What is the Fundamental Theorem of Algebra?

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The Fundamental Theorem of Algebra states that every non-constant polynomial equation of degree n has exactly n roots in the complex number system.

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How many roots does a polynomial of degree 4 have according to the Fundamental Theorem of Algebra?

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A polynomial of degree 4 has exactly 4 roots, counting multiplicities.

Front

What are the zeros of the polynomial g(x) = 2x^4 - 19x^2 + 46x + 7x - 60?

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The zeros are -1, 4, 5, 3/2.

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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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What is the relationship between factors and zeros of a polynomial?

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If x = r is a zero of a polynomial, then (x - r) is a factor of that polynomial.