Rational Zero Theorem, Descartes Rule of Signs, Irrational Zeros

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What is the Rational Zero Theorem?

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The Rational Zero Theorem states that if a polynomial has rational roots, they can be expressed as p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

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How do you apply the Rational Zero Theorem to find possible rational zeros?

Back

List all factors of the constant term and the leading coefficient, then form all possible fractions p/q.

Tags

CCSS.HSA.APR.B.2

Front

What is Descartes' Rule of Signs?

Back

Descartes' Rule of Signs is a method for determining the number of positive and negative real roots of a polynomial based on the number of sign changes in the polynomial's coefficients.

Front

How do you determine the number of positive roots using Descartes' Rule of Signs?

Back

Count the number of sign changes in f(x). The number of positive roots is equal to the number of sign changes or less by an even number.

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How do you determine the number of negative roots using Descartes' Rule of Signs?

Back

Count the number of sign changes in f(-x). The number of negative roots is equal to the number of sign changes or less by an even number.

Front

What is a rational zero?

Back

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.

Tags

CCSS.HSA.APR.B.2

Front

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

Back

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

Tags

CCSS.HSA.APR.B.2

Front

What does it mean if a polynomial has no rational zeros?

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It means that the polynomial does not have any solutions that can be expressed as fractions of integers.

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How can you verify if a rational zero is indeed a root of the polynomial?

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

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

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Finding rational zeros helps in factoring the polynomial, simplifying it, and finding all possible roots.