Complex Numbers and Polynomial Zeros

This video tutorial introduces 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 tutorial covers finding zeros, using synthetic division, and applying the quadratic formula to 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 real 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 distinct zeros.

What is the form of a complex number?

• a + b

• a - b

• a + bi

• a - bi

How can a polynomial function be expressed using its zeros?

• As a sum of linear factors.

• As a product of linear factors.

• As a difference of linear factors.

• As a quotient of linear factors.

In the first example, what is the zero of the polynomial function?

• 0

• 5

• -5

• 10

What method is used to divide the polynomial by x - 5 in the first example?

• Long division

• Synthetic division

• Factorization

• Partial fraction decomposition

What are the zeros of the polynomial x^2 + 16?

• 4i and -4i

• 4 and -4

• 8i and -8i

• 8 and -8

In the second example, what is the behavior of the graph at the x-intercept (2,0)?

• It oscillates.

• It remains constant.

• It touches and turns back.

• It crosses the x-axis.

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

• 4

• 3

• 2

• 1

Which formula is used to find the zeros of x^2 - 4x + 13?

• Binomial theorem

• Factor theorem

• Quadratic formula

• Pythagorean theorem

What are the complex zeros of the polynomial x^2 - 4x + 13?

• 2 + 3i and 2 - 3i

• 3 + 2i and 3 - 2i

• 5 + 4i and 5 - 4i

• 4 + 5i and 4 - 5i