Fundamental Theorem of Algebra

Front

What does the Fundamental Theorem of Algebra state?

Back

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

Front

What is a polynomial?

Back

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.

Front

Define the degree of a polynomial.

Back

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

Front

How many roots does a polynomial of degree 4 have?

Back

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

Front

What is a complex root?

Back

A complex root is a solution to a polynomial equation that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.

Front

What is the significance of complex roots in polynomials?

Back

Complex roots occur in conjugate pairs, meaning if a + bi is a root, then a - bi is also a root.

Front

What is the relationship between the number of x-intercepts and the degree of a polynomial?

Back

The maximum number of x-intercepts of a polynomial is equal to its degree.

Front

What is the maximum number of positive roots for a polynomial?

Back

The maximum number of positive roots is determined by the number of sign changes in the polynomial's coefficients.

Front

What does it mean for a polynomial to have multiple roots?

Back

A polynomial has multiple roots when a particular root occurs more than once, affecting the graph's behavior at that point.

Front

What is the Fundamental Theorem of Algebra's implication for real coefficients?

Back

If a polynomial has real coefficients, any non-real roots must occur in conjugate pairs.