Q: Which of the following is the correct application of the Linear Factorization Theorem for a cubic polynomial $$f(x)$$ with zeros $$2$$, $$-1$$, and $$3i$$?

$$f(x) = a(x-2)(x+1)(x-3i)$$,\ a \neq 0$$ (but 3i$ must have its conjugate as a zero for real coefficients)

Question 1

Given the polynomial $$f(x) = 4x^3$ - 3x^2$ + 2x - 3$$, use the Rational Zero Theorem to list all possible rational zeros.

Accepted Answer

$$\pm1, \pm3, \pm\frac{1}{4}, \pm\frac{3}{4}$$

Question 2

Which of the following is a property of the roots of a polynomial equation with real coefficients?

Accepted Answer

Complex roots always occur in conjugate pairs.

Question 3

If $$f(x) = x^3$ - 2x^2$ - x + 2$$, which of the following is a zero of $$f(x)$$?

Accepted Answer

$$x = 2$$

Question 4

A third-degree polynomial with real coefficients has zeros at $$-3$$ and $$i. $$What must be the third zero?

Accepted Answer

$$-i$$

Question 5

Given $$f(x) = x^4$ - 5x^3$ + 6x^2$ - x + 2$$, how many positive real zeros are possible according to Descartes’ Rule of Signs?

Accepted Answer

4, 2, or 0

Question 6

Which statement best describes the Fundamental Theorem of Algebra?

Accepted Answer

Every polynomial equation of degree $$n$$ has at least one complex root.

Question 7

If $$f(x) = x^3$ + 2x^2$ - x - 2$$, and you know $$x = 1 is a $$root, what is the depressed quadratic after dividing by $$(x-1)$$?

Accepted Answer

$$x^2 + 3x + 2$

Question 8

Which of the following is the correct application of the Linear Factorization Theorem for a cubic polynomial $$f(x)$$ with zeros $$2$$, $$-1$$, and $$3i$$?

Accepted Answer

$$f(x) = a(x-2)(x+1)(x-3i)$$,\ a 
eq 0$$ (but 3i$ must have its conjugate as a zero for real coefficients)

Question 9

A chemical reaction’s rate is modeled by $$f(x) = 2x^3$ - 5x^2$ + x - 2. $$Using the Rational Zero Theorem, which of the following is NOT a possible rational zero?

Accepted Answer

$$\frac{3}{2}$$

Question 10

If $$f(x) = x^3$ - 4x^2$ + 5x - 2$$, and $$x = 1 is a $$root, what are the remaining roots?

Accepted Answer

$$x = 1, x = 1, x = 2 ($$with $$x = 1 as a $$repeated root)

Question 11

A polynomial $$f(x) of $$degree 4 has real coefficients and exactly two sign changes in $$f(x). $$According to Descartes’ Rule of Signs, how many positive real zeros could $$f(x)$$ have?

Accepted Answer

2 or 0

Question 12

Suppose a polynomial $$f(x)$$ has degree 5 and real coefficients. If $$2 + i is a $$root, which of the following must also be a root?

Accepted Answer

$$2 - i$$

Question 13

A population model is given by $$f(x) = x^3$ - 6x^2$ + 11x - 6. $$Which of the following is NOT a zero of $$f(x)$$?

Accepted Answer

$$x = 4$$

Question 14

If $$f(x) = x^4$ + 2x^3$ - 7x^2$ - 8x + 12$$, how many negative real zeros are possible according to Descartes’ Rule of Signs?

Accepted Answer

2 or 0

Question 15

Which of the following best describes the Linear Factorization Theorem?

Accepted Answer

Every nth-degree polynomial can be written as a product of n linear factors over the complex numbers.

Zeros of Polynomial Functions and Related Theorems

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