Relate Factors and Zeros Using the Factor Theorem

This video tutorial explores the relationship between factors and zeros using the factor theorem. It begins with an introduction to the concepts and then delves into the remainder theorem, providing examples to illustrate its application. The video distinguishes between polynomial expressions and functions, and explains the factor theorem in detail. Finally, it demonstrates how to apply the factor theorem to determine factors and zeros, using graphs and examples to solidify understanding.

What does the remainder theorem state about the remainder when a polynomial is divided by x-a?

• The remainder is equal to the degree of the polynomial.

• The remainder is always zero.

• The remainder is equal to p(a).

• The remainder is equal to a constant.

What is significant about obtaining a remainder of zero when dividing a polynomial?

• It means the divisor is not a factor.

• It indicates the polynomial is constant.

• It shows the divisor is a factor.

• It implies the polynomial has no roots.

How can polynomial expressions be distinguished from polynomial functions?

• Expressions have multiple variables, functions have one.

• Expressions are in one variable, functions involve multiple variables.

• Expressions are equations, functions are inequalities.

• Expressions are always linear, functions are quadratic.

According to the factor theorem, when is x-a a factor of a polynomial p(x)?

• When p(a) is greater than zero.

• When p(a) is less than zero.

• When p(a) equals zero.

• When p(a) is a constant.

What does the factor theorem imply if a is a zero of p(x)?

• x-a is not a factor of p(x).

• x-a is a factor of p(x).

• p(x) has no x-intercepts.

• p(x) is a constant function.

How can the factor theorem be used to determine if x-3 is a factor of a polynomial?

• By finding the degree of the polynomial.

• By checking if 3 is a zero of the polynomial.

• By performing synthetic division.

• By evaluating the polynomial at zero.

What can be inferred from the x-intercepts of a graph regarding the polynomial's factors?

• The x-intercepts show the polynomial's leading coefficient.

• The x-intercepts help identify the polynomial's factors.

• The x-intercepts indicate the polynomial's degree.

• The x-intercepts are unrelated to the factors.