Complex Numbers and Polynomial Functions

This video tutorial covers the fundamental theorem of algebra, explaining that a polynomial function of degree n has at least one complex zero. It discusses the concept of complex numbers and their role in polynomial equations. The tutorial demonstrates how to determine the number of real and imaginary zeros graphically using a calculator. It introduces the conjugate zeros theorem, which states that complex zeros come in conjugate pairs. The video also covers factoring polynomials with complex zeros and solving polynomial equations with complex solutions, providing examples throughout.

What does the fundamental theorem of algebra state about polynomial functions?

• They have only real zeros.

• They have no zeros.

• They have at least one complex zero.

• They have at least one real zero.

Which of the following is a form of a complex number?

• a - b

• a + bi

• a * b

• a + b

According to the number of zeros theorem, how many distinct zeros can a polynomial of degree n have?

• No more than n-1

• Exactly n

• At most n

• At least n

When graphing a polynomial, what does an x-intercept represent?

• A complex zero

• A real zero

• An imaginary zero

• A zero of multiplicity two

What does the conjugate zeros theorem state about complex zeros?

• They are always imaginary.

• They are always real.

• They appear as single entities.

• They always come in pairs.

If 2 + 3i is a zero of a polynomial, what is its conjugate?

• 2 - 3i

• -2 - 3i

• 2 + 3i

• -2 + 3i

What is the first step in factoring a polynomial with complex zeros?

• Use synthetic division.

• Graph the polynomial.

• Use the quadratic formula.

• Find the greatest common factor.

When solving a polynomial equation with complex solutions, what is a common first step?

• Divide both sides by a variable.

• Multiply both sides by a constant.

• Set the equation equal to zero.

• Add a constant to both sides.