Polynomial Functions and Their Properties

The video tutorial covers the remainder and factor theorems, explaining how to determine if a polynomial has a specific factor and how to find the remainder when dividing polynomials. It provides examples using synthetic division and demonstrates how to factor a polynomial into linear factors. The tutorial also explains the concept of real zeros and their graphical representation on the x-axis.

What does the Remainder Theorem state about the remainder when a polynomial f(x) is divided by x - k?

• The remainder is equal to x.

• The remainder is always zero.

• The remainder is equal to f(k).

• The remainder is equal to k.

According to the Factor Theorem, when is x - k a factor of a polynomial f(x)?

• When f(k) is not equal to zero.

• When f(k) is equal to zero.

• When f(k) is less than zero.

• When f(k) is greater than zero.

What does it mean for a number to be a zero of a function?

• The function value is undefined at that number.

• The function value is zero at that number.

• The function value is negative at that number.

• The function value is positive at that number.

In the example problem, what is the value of f(1) for the polynomial 2x^3 - 3x^2 - 5x + 6?

• 0

• -1

• 2

• 1

What method is used to divide x - 1 into the polynomial f(x) in the example problem?

• Factorization

• Polynomial division

• Synthetic division

• Long division

What is the quotient when 2x^3 - 3x^2 - 5x + 6 is divided by x - 1 using synthetic division?

• 2x^2 - x - 6

• 2x^2 + x - 6

• 2x^2 - x + 6

• 2x^2 + x + 6

What are the linear factors of the polynomial 2x^3 - 3x^2 - 5x + 6?

• x - 1, 2x + 3, x - 2

• x + 1, 2x + 3, x + 2

• x + 1, 2x - 3, x + 2

• x - 1, 2x - 3, x + 2

What are the real zeros of the polynomial 2x^3 - 3x^2 - 5x + 6?

• 1, -3/2, 2

• 1, 3/2, -2

• 1, -3/2, -2

• 1, 3/2, 2

What is the purpose of writing a polynomial as a product of linear factors?

• To change the polynomial's coefficients.

• To simplify the polynomial.

• To identify the real zeros of the polynomial.

• To increase the degree of the polynomial.

What is the graphical significance of the real zeros of a polynomial?

• They are the points where the graph has a maximum.

• They are the points where the graph has a minimum.

• They are the points where the graph intersects the y-axis.

• They are the points where the graph intersects the x-axis.