Fundamental Theorem of Algebra

Front

What is the Fundamental Theorem of Algebra?

Back

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

Front

Define a polynomial's degree.

Back

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

Front

What are the possible types of roots for a polynomial of degree 6?

Back

A polynomial of degree 6 can have 0, 2, 4, or 6 real roots, and the remaining roots will be imaginary, occurring in conjugate pairs.

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What is a zero of a polynomial?

Back

A zero of a polynomial is a value of x that makes the polynomial equal to zero.

Front

How do you find the zeros of a polynomial?

Back

To find the zeros of a polynomial, set the polynomial equal to zero and solve for x.

Front

What is multiplicity in relation to polynomial roots?

Back

Multiplicity refers to the number of times a particular root appears in the factorization of a polynomial.

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If a polynomial has a root of 3 with multiplicity 2, what does this mean?

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It means that (x - 3)² is a factor of the polynomial, indicating that the root 3 is repeated twice.

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What is the relationship between the coefficients and roots of a polynomial?

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The coefficients of a polynomial can be expressed in terms of its roots using Vieta's formulas.

Front

Write a polynomial in standard form given the roots 2 and -3i.

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The polynomial is: x³ + 3ix² + 2x + 6i.

Front

What is the significance of complex roots in polynomials?

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Complex roots occur in conjugate pairs, meaning if a polynomial has a complex root a + bi, it must also have a root a - bi.