Polynomial Functions and Theorems

The video tutorial explains the complex conjugate root theorem, which states that if a polynomial with real coefficients has a complex root, its conjugate is also a root. The tutorial provides an example polynomial and demonstrates how to apply the theorem to find roots. It further explores solving an exam-type question using the theorem and the factor theorem to find remaining roots. The tutorial concludes with a long division of polynomials to determine the quotient polynomial, summarizing the application of the theorem.

What does the complex conjugate root theorem state about the roots of polynomial functions with real coefficients?

• All roots are real numbers.

• There are no complex roots.

• Complex roots appear in pairs.

• Complex roots are always imaginary.

If a polynomial has a complex root Z, what must also be true according to the theorem?

• Z is not a valid root.

• The complex conjugate of Z is also a root.

• The polynomial has no other roots.

• Z is the only root.

In the example provided, what is the complex conjugate of the root 1 - 2i?

• -1 - 2i

• 1 - 2i

• -1 + 2i

• 1 + 2i

What is the first step in solving the exam question using the complex conjugate root theorem?

• Ignoring the complex roots.

• Using the factor theorem.

• Finding the derivative.

• Guessing the roots.

What is the polynomial function given in the exam question?

• f(x) = 3x^2 - x + 4

• f(x) = x^3 - 2x + 1

• f(x) = 2x^3 - 5x^2 + 17x - 13

• f(x) = x^2 + 3x + 2

What method is used to find the remaining root after identifying the complex roots?

• Graphical method.

• Substitution method.

• Long division of polynomials.

• Trial and error.

What is the result of dividing the polynomial by the product of the factors (x - Z) and (x - Z*)?

• Zero.

• A linear polynomial.

• A constant.

• A quadratic polynomial.

What is the remaining root of the polynomial function after applying the theorem?

• x = 0

• x = 2

• x = 1

• x = -1

What does the factor theorem help determine in the context of the exam question?

• The factors of the polynomial.

• The coefficients of the polynomial.

• The degree of the polynomial.

• The derivative of the polynomial.

What is the significance of the remainder being zero in polynomial division?

• The divisor is not a factor.

• The polynomial is irreducible.

• The divisor is a factor.

• The division was incorrect.