Apply the Fundamental Theorem of Algebra

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What does the Fundamental Theorem of Algebra state?

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Every non-constant polynomial equation of degree n has exactly n roots in the complex number system.

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What is a rational root?

Back

A rational root is a solution to a polynomial equation that can be expressed as a fraction of two integers.

Front

How can you determine the possible rational roots of a polynomial?

Back

Use the Rational Root Theorem, which states that any rational solution, expressed as p/q, has p as a factor of the constant term and q as a factor of the leading coefficient.

Front

What is the degree of a polynomial?

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The degree of a polynomial is the highest power of the variable in the polynomial.

Front

How does the degree of a polynomial relate to the number of roots?

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The degree of the polynomial indicates the maximum number of roots it can have, including complex and repeated roots.

Front

What is Descartes' Rule of Signs?

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Descartes' Rule of Signs helps to determine the number of positive and negative real roots of a polynomial by counting the sign changes in the polynomial's coefficients.

Front

If a polynomial has a degree of 4, what is the maximum number of real roots it can have?

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It can have up to 4 real roots.

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What does it mean if a polynomial has complex roots?

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Complex roots occur in conjugate pairs and are solutions that cannot be represented on the real number line.

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How can you find the number of positive roots of a polynomial?

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Count the number of sign changes in the polynomial's coefficients for f(x).

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How can you find the number of negative roots of a polynomial?

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Count the number of sign changes in the polynomial's coefficients for f(-x).