How to Find All of the Zeros of a Polynomial by Factoring, Including Imaginary

The video tutorial covers solving polynomial functions, focusing on factoring and the zero product property. It explains the use of synthetic division and the rational zero test. The tutorial introduces imaginary numbers when dealing with square roots of negative numbers and discusses the importance of writing linear factors correctly. The fundamental theorem of algebra is explored, emphasizing the types of zeros: rational, irrational, and complex, and their relationship to the degree of the polynomial.

What is the first step in solving a polynomial function that is already in factored form?

• Expand the polynomial

• Apply the rational zero test

• Perform synthetic division

• Use the zero product property

When factoring a polynomial, what should you do if you encounter a negative square root?

• Factor it out as a real number

• Convert it to a positive number

• Ignore it

• Use the imaginary number system

Why is it important to write the imaginary unit 'i' in front of the square root?

• To ensure the square root is positive

• To follow standard mathematical notation

• To make calculations easier

• To avoid confusion with real numbers

What is the significance of the multiplicity of a factor in a polynomial?

• It determines the polynomial's degree

• It indicates how many times a zero is repeated

• It shows the polynomial's leading coefficient

• It affects the polynomial's constant term

What does the Fundamental Theorem of Algebra state about the degree of a polynomial?

• It shows the polynomial's minimum value

• It equals the number of zeros or solutions

• It determines the number of terms in the polynomial

• It indicates the polynomial's maximum value

How do complex zeros typically appear in a polynomial?

• As single zeros

• In pairs

• As rational numbers

• As irrational numbers

What types of zeros can a polynomial have according to the Fundamental Theorem of Algebra?

• Only complex zeros

• Only rational zeros

• Rational, irrational, or complex zeros

• Only irrational zeros