4.6 The Fundamental Theorem of Algebra

Front

What is the Fundamental Theorem of Algebra?

Back

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

Front

How many roots can a polynomial of degree 6 have?

Back

A polynomial of degree 6 can have 6 roots in total, which may include real and complex (imaginary) roots.

Front

If a polynomial has 4 real roots and 2 complex roots, what is its degree?

Back

The degree of the polynomial is 6, as the total number of roots (real + complex) must equal the degree.

Front

What is a polynomial function of least degree with integral coefficients that has the zeros 1, -2, and 3?

Back

The polynomial is (x - 1)(x + 2)(x - 3) = x^3 - 2x^2 - 5x + 6.

Front

What is the relationship between the degree of a polynomial and the number of its roots?

Back

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

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

If a polynomial has a complex root of 2 + 3i, what is its conjugate root?

Back

The conjugate root is 2 - 3i.

Front

Write a cubic polynomial in standard form with roots -4, 2 + 3i, and 2 - 3i.

Back

The polynomial is (x + 4)(x - (2 + 3i))(x - (2 - 3i)) = x^3 - 6x^2 + 13x + 48.

Front

What is the degree of a polynomial with 5 real roots?

Back

The degree of the polynomial is 5.

Front

Can a polynomial of degree 4 have 3 imaginary roots?

Back

No, a polynomial of degree 4 can have at most 4 roots, and if it has 3 imaginary roots, it must have 1 real root.