Question 1

Given the polynomial $$f(x) = 4x^5$ + x^4$ - 12x^3$ - x^2$, use the Rational Zero Theorem to list all possible rational zeros.

Accepted Answer

$$\pm1, \pm2, \pm4, \pm3, \pm6, \pm12, \pm\frac{1}{4}, \pm\frac{3}{4}, \pm\frac{6}{4}, \pm\frac{12}{4}$$

Question 2

If $$f(x) = 5x^3$ + 2x^2$ - x - 3$$, which of the following is NOT a possible rational zero according to the Rational Zero Theorem?

Accepted Answer

$$\frac{1}{2}$$

Question 3

A polynomial with real coefficients has a root at $$2 + 5i. $$Which of the following must also be a root?

Accepted Answer

$$2 - 5i$$

Question 4

How many roots (counting multiplicity) does the equation $$x^4$$ - $$6x^3$$ + $$22x^2 - 30x + 13 = 0$ have?

Accepted Answer

4

Question 5

Given $$f(x) = x^4$ - 6x^3$ + 22x^2$ - 30x + 13$$, use Descartes’ Rule of Signs to determine the possible number of positive real zeros.

Accepted Answer

2 or 0

Question 6

Which statement best describes the Fundamental Theorem of Algebra?

Accepted Answer

Every polynomial equation of degree $$n$$ with complex coefficients has exactly $$n$$ complex roots, counting multiplicities.

Question 7

If $$f(x) = x^4$ - 6x^3$ + 22x^2$ - 30x + 13$$, what is the possible number of negative real zeros according to Descartes’ Rule of Signs?

Accepted Answer

2 or 0

Question 8

A third-degree polynomial $$f(x)$$ with real coefficients has zeros at $$-3$$ and $$i$$, and $$f(1) = 8. $$Which of the following is the correct form for $$f(x)$$?

Accepted Answer

$$f(x) = a(x + 3)(x - i)(x + i)$$, where $$a is a $$real number$

Question 9

Suppose a chemical reaction is modeled by the polynomial $$f(x) = x^3$ - 2x^2$ - 5x + 6. $$Which of the following is a rational zero of $$f(x)$$?

Accepted Answer

$$-1$$

Question 10

A polynomial $$f(x) of $$degree 4 has real coefficients and exactly two distinct real roots, one of which is a double root. Which of the following could be the complete set of roots?

Accepted Answer

$$1, 1, 2, 2 + 0i$$

Question 11

If $$f(x) = x^4$ - 6x^3$ + 22x^2$ - 30x + 13$$ has a root at $$x = 1 + 2i$$, which of the following must also be a root?

Accepted Answer

$$1 - 2i$$

Question 12

A pharmaceutical company models the concentration of a drug in the bloodstream by $$C(t) = t^3$ - 4t^2$ + 5t - 2. $$Which of the following is a possible rational zero?

Accepted Answer

$$-2$$

Question 13

Which of the following statements is TRUE about the roots of a polynomial equation with real coefficients?

Accepted Answer

If $$a + bi is a $$root, then $$a - bi is $$also a root.

Question 14

A polynomial $$f(x)$$ has degree 4 and exactly two non-real complex roots. Which of the following is a possible set of roots?

Accepted Answer

$$1, 2, i, -i$$

Question 15

If a polynomial $$f(x)$$ has degree 5, what is the maximum number of positive real zeros it can have?

Accepted Answer

5

Question 16

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

Accepted Answer

$$-1$$

Question 17

Which of the following is the correct application of the Linear Factorization Theorem for a polynomial $$f(x) of $$degree $$n$$ with real coefficients?

Accepted Answer

$$f(x) = a(x - r_1$)(x - r_2$)\cdots(x - r_n)$$, where $$r_i$$ are the complex roots$

Question 18

A polynomial $$f(x)$$ has degree 4 and exactly two distinct real roots, both with multiplicity 2. Which of the following could be $$f(x)$$?

Accepted Answer

$$f(x) = (x - 1)^2$(x - 2)^2$

Question 19

If a polynomial $$f(x)$$ has degree 3 and all its roots are real and distinct, which of the following could be its graph?

Accepted Answer

A cubic curve crossing the x-axis at three points

Zeros of Polynomial Functions and Related Theorems

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