Polynomial Functions and Their Zeros

The video tutorial explains how to find all zeros of a polynomial function and express it as a product of linear factors. It begins with a review of the Complex Factorization Theorem, then demonstrates finding real rational zeros using graphing and the Rational Roots Theorem. The tutorial proceeds with synthetic division to simplify the polynomial and applies the quadratic formula to find remaining zeros. Finally, it summarizes the process and presents the polynomial as a product of linear factors.

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

• A polynomial of degree n has n real zeros.

• A polynomial of degree n has n distinct zeros.

• A polynomial of degree n has n rational zeros.

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

According to the Rational Roots Theorem, where do the potential rational roots of a polynomial come from?

• The ratio of the factors of the leading coefficient to the factors of the constant term.

• The sum of the factors of the constant term and the leading coefficient.

• The difference between the factors of the constant term and the leading coefficient.

• The ratio of the factors of the constant term to the factors of the leading coefficient.

What is the significance of finding an x-intercept on the graph of a polynomial function?

• It indicates a real zero.

• It indicates a rational zero.

• It indicates an irrational zero.

• It indicates a complex zero.

What is the first step in synthetic division when dividing a polynomial by a factor?

• List the coefficients of the polynomial.

• Multiply the leading coefficient by the divisor.

• Add the coefficients of the polynomial.

• Subtract the divisor from the polynomial.

What is the remainder when a polynomial is exactly divisible by a factor?

• The remainder is equal to the dividend.

• The remainder is zero.

• The remainder is equal to the divisor.

• The remainder is one.

Which formula is used to find the zeros of a quadratic factor that cannot be factored?

• The Quadratic Formula

• The Linear Factorization Theorem

• The Complex Factorization Theorem

• The Rational Roots Theorem

What are the values of A, B, and C in the quadratic formula for the polynomial x^2 - 6x + 6?

• A = -6, B = 1, C = 6

• A = 6, B = 1, C = -6

• A = 1, B = -6, C = 6

• A = 1, B = 6, C = -6

What are the two remaining zeros of the polynomial after using the quadratic formula?

• x = 3 ± √3

• x = 3 ± √2

• x = 2 ± √3

• x = 2 ± √2

How is the polynomial function finally expressed after finding all zeros?

• As a product of four linear factors

• As a product of three linear factors

• As a product of two linear factors

• As a product of one linear factor

Which of the following describes the nature of the zeros found in the polynomial?

• All zeros are real and irrational.

• One zero is real and rational, two are real and irrational.

• All zeros are complex.

• All zeros are real and rational.