Complex Zeros and Polynomial Factorization

The video tutorial explains how to find all zeros of a degree 3 polynomial and express it as a product of linear factors. It covers the complex factorization theorem, graphing to find real zeros, and using synthetic division to identify factors. The tutorial also demonstrates solving for imaginary zeros using square roots and concludes with the complete factorization of the polynomial.

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

• There are n distinct zeros.

• There are n complex zeros, counting multiplicity.

• There are n imaginary zeros.

• There are n real zeros.

How can real rational zeros of a polynomial be found using a graph?

• By counting the number of peaks in the graph.

• By finding the points where the graph touches the x-axis.

• By identifying the y-intercepts.

• By looking for the highest point on the graph.

What does it mean if x = -4 is a zero of the polynomial?

• x - 4 is a factor of the polynomial.

• x + 4 is a factor of the polynomial.

• The polynomial has no real zeros.

• The polynomial is degree 4.

What is the purpose of using synthetic division in this context?

• To calculate the derivative of the polynomial.

• To find the remainder of the polynomial.

• To verify if a given factor is correct and find the quotient.

• To determine the degree of the polynomial.

What is the result of dividing the polynomial by x + 4 using synthetic division?

• x^2 - 4

• x^2 - 9

• x^2 + 4

• x^2 + 9

How can the zeros of the quadratic factor x^2 + 9 be found?

• By factoring the quadratic.

• By completing the square.

• By taking the square root of both sides.

• By using the quadratic formula.

What are the zeros of the quadratic factor x^2 + 9?

• 3i and -3i

• 3 and -3

• 9 and -9

• i and -i

What is the complete factorization of the polynomial P(x)?

• x - 4, x + 3, x - 3

• x - 4, x + 3i, x - 3i

• x + 4, x - 3i, x + 3i

• x + 4, x - 3, x + 3

What does each linear factor in the factorization represent?

• A zero of the polynomial.

• A maximum point of the polynomial.

• A turning point of the polynomial.

• A minimum point of the polynomial.

What is the significance of the imaginary zeros in the factorization?

• They indicate the polynomial has no real solutions.

• They show the polynomial is not factorable.

• They represent complex solutions to the polynomial.

• They are irrelevant to the factorization.