Polynomial Functions and Their Properties

The video tutorial explains the Fundamental Theorem of Algebra, which states that every polynomial function with complex coefficients has at least one complex zero. It also covers the corollary that a polynomial of degree n has exactly n zeros. The tutorial provides examples of three polynomial functions, demonstrating how to find their degrees and zeros through inspection and factoring. The first example involves a cubic polynomial, the second a quadratic polynomial with imaginary zeros, and the third a quartic polynomial.

What does the Fundamental Theorem of Algebra state about polynomial functions?

• They have at least one real zero.

• They have at least one complex zero.

• They have only real zeros.

• They have no zeros.

According to the corollary of the Fundamental Theorem of Algebra, what is true about the degree of a polynomial?

• It is always greater than the number of zeros.

• It is always less than the number of zeros.

• It is equal to the number of zeros.

• It has no relation to the number of zeros.

What is the degree of the polynomial function f(x) = 4x^3 - 12x^2 + 9x?

• 4

• 3

• 1

• 2

How many zeros does the polynomial f(x) = 4x^3 - 12x^2 + 9x have?

• 4

• 3

• 2

• 1

What is the degree of the polynomial function f(x) = x^2 + 25?

• 2

• 0

• 1

• 3

How many zeros does the polynomial f(x) = x^2 + 25 have?

• 4

• 1

• 3

• 2

What is the degree of the polynomial function f(x) = x^4 - 10x^2 + 9?

• 5

• 4

• 3

• 2

How many zeros does the polynomial f(x) = x^4 - 10x^2 + 9 have?

• 3

• 5

• 4

• 2

What is the common factor in the polynomial 4x^3 - 12x^2 + 9x?

• x

• 2x

• 3x

• 4x

What is the result of factoring 4x^3 - 12x^2 + 9x completely?

• x(2x + 3)^2

• x(4x - 3)^2

• x(2x - 3)^2

• x(4x + 3)^2