Zeros and Factors of Polynomials

The video tutorial explains how to find all zeros of a polynomial function, given a complex zero, and how to express the polynomial as a product of linear factors. It covers the complex factorization theorem, graphing to identify real rational intercepts, and using synthetic division to find remaining factors. The tutorial concludes with finding the final linear factor using complex division.

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

• There are as many real zeros as the degree of the polynomial.

• There are as many complex zeros as the degree of the polynomial, counting multiplicity.

• There are twice as many complex zeros as the degree of the polynomial.

• There are no complex zeros in a polynomial function.

If 4i is a zero of a polynomial, what can be inferred about its conjugate?

• The conjugate is not a zero.

• The conjugate is 4i.

• The conjugate is a real number.

• The conjugate, -4i, is also a zero.

How can real zeros be identified from a polynomial graph?

• By checking where the graph is horizontal.

• By finding the y-intercepts.

• By identifying the x-intercepts.

• By looking for the highest point on the graph.

What is the purpose of using synthetic division in this context?

• To find the degree of the polynomial.

• To find linear factors and zeros of the polynomial.

• To determine the leading coefficient.

• To graph the polynomial function.

What is the result of dividing a polynomial by one of its linear factors?

• A constant value.

• A polynomial of lower degree.

• A polynomial of the same degree.

• A polynomial of higher degree.

When performing synthetic division with a complex number, what should the remainder be if the divisor is a factor?

• The remainder should be 1.

• The remainder should be zero.

• The remainder should be negative.

• The remainder should be equal to the divisor.

What is the last zero found in the polynomial function discussed?

• 2/3

• -4i

• 4i

• -1

What is the relationship between zeros and linear factors in a polynomial?

• Each zero corresponds to a linear factor.

• Zeros and linear factors are unrelated.

• Zeros are half the number of linear factors.

• Linear factors are twice the number of zeros.

How many linear factors does a degree 5 polynomial have?

• 6

• 3

• 4

• 5

What method can be used to find the remaining zero after identifying some zeros?

• Graphing the polynomial again.

• Using the Rational Roots Theorem.

• Guessing the remaining zero.

• Performing division with known factors.