Complex Zeros and Polynomial Factorization

This video tutorial covers the complex factorization theorem, explaining how a polynomial function of degree n with complex coefficients has exactly n complex zeros, considering multiplicity. It demonstrates the theorem through two examples: a degree three polynomial and a degree four polynomial. The video uses synthetic division and the quadratic formula to find zeros and express polynomials as products of linear factors.

What does the Complex Factorization Theorem state about the zeros of a polynomial?

• A polynomial of degree n has n distinct zeros.

• A polynomial of degree n has n imaginary zeros.

• A polynomial of degree n has n complex zeros, counting multiplicity.

• A polynomial of degree n has n real zeros.

How can a polynomial function be expressed using its complex zeros?

• As a quotient of quadratic factors.

• As a sum of quadratic factors.

• As a product of linear factors.

• As a difference of linear factors.

In the first example, what method is used to find the quadratic factor after identifying a zero?

• Synthetic division

• Completing the square

• Factoring by grouping

• Long division

What is the zero of the polynomial function in the first example?

• x = 0

• x = -5

• x = 5

• x = 4

What are the remaining zeros of the polynomial in the first example after finding the real zero?

• x = 4 and x = -4

• x = 4i and x = -4i

• x = 5i and x = -5i

• x = 3i and x = -3i

In the second example, what indicates that a zero has even multiplicity?

• The graph touches the x-axis and turns back.

• The graph crosses the x-axis.

• The graph does not touch the x-axis.

• The graph has a vertical asymptote.

What method is used to find the remaining linear factors in the second example?

• Graphical method

• Long division

• Quadratic formula

• Completing the square

What are the complex zeros found using the quadratic formula in the second example?

• x = 3 ± 2i

• x = 2 ± 3i

• x = 3 ± 4i

• x = 4 ± 3i

What is the multiplicity of the zero x = 2 in the second example?

• Multiplicity 3

• Multiplicity 4

• Multiplicity 1

• Multiplicity 2

What is the significance of complex zeros in polynomial factorization?

• They only appear in quadratic polynomials.

• They are not considered in factorization.

• They help express the polynomial as a product of linear factors.

• They are always real numbers.