Question 1

Given the polynomial $$f(x) = 2x^3$ + 5x^2$ - 6x - 6$$, use the Rational Zero Theorem to list all possible rational zeros.

Accepted Answer

$$\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{6}{2}$$

Question 2

For the polynomial $$f(x) = x^4$ - 2x^2$ - 16x + 15$$, how many possible positive real zeros are there according to Descartes' Rule of Signs?

Accepted Answer

2 or 0

Question 3

Which of the following is NOT a possible rational zero of $$f(x) = 3x^5$ - 4x^4$ + 12$$?

Accepted Answer

$$\frac{4}{3}$$

Question 4

If $$f(x) = x^3$ + 8x^2$ + 11x - 20$$, and you know that $$x = 1 is a $$zero, what is the depressed quadratic factor after dividing by $$(x-1)$$?

Accepted Answer

$$x^2 + 9x + 20$

Question 5

Given $$f(x) = x^3$ + 5x^2$ - x - 2$$, which of the following is a possible rational zero according to the Rational Zero Theorem?

Accepted Answer

$$-2$$

Question 6

A cubic polynomial with integer coefficients has zeros at $$-5$$ and $$4 + 3i. $$What is the third zero?

Accepted Answer

$$4 - 3i$$

Question 7

If $$f(x) = x^4$ - 2x^2$ - 16x + 15$$, use the Rational Zero Theorem to list all possible rational zeros.

Accepted Answer

$$\pm1, \pm3, \pm5, \pm15, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{5}{2}, \pm\frac{15}{2}$$

Question 8

Suppose $$f(x) = x^3$ - 2x^2$ - 5x + 6. $$According to Descartes' Rule of Signs, how many negative real zeros can $$f(x)$$ have?

Accepted Answer

2 or 0

Question 9

A chemical process is modeled by $$f(x) = x^3$ - 2x^2$ - 5x + 6. If$$ $$x$$ represents the concentration of a reactant, which of the following values is a rational zero that could represent a physically meaningful (positive) concentration?

Accepted Answer

$$3$$

Question 10

Given $$f(x) = x^3$ + 5x^2$ - x - 2$$, and you know $$x = -2 is a $$zero, what is the resulting quadratic factor after division?

Accepted Answer

$$x^2 + 3x + 1$

Question 11

A polynomial $$f(x)$$ has integer coefficients and $$\frac{3}{2} is a $$zero. According to the Rational Zero Theorem, which of the following must be true?

Accepted Answer

2 divides the leading coefficient and 3 divides the constant term

Question 12

If $$f(x) = x^3$ + 8x^2$ + 11x - 20$$ and you have found $$x = 1 is a $$zero, what are the remaining zeros?

Accepted Answer

$$-4$$ and $$-5$$

Question 13

A real-world scenario: The profit $$P(x)$$ from selling $$x$$ units of a product is modeled by $$P(x) = x^3$ - 6x^2$ + 11x - 6. $$What is the minimum number of units that must be sold to break even (i.e., $$P(x) = 0$$)?

Accepted Answer

$$1$$

Question 14

Given $$f(x) = x^3$ - 2x^2$ - 5x + 6$$, which of the following is NOT a possible rational zero?

Accepted Answer

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

Question 15

If a polynomial $$f(x)$$ has integer coefficients and $$2 + i is a $$zero, which of the following must also be a zero?

Accepted Answer

$$2 - i$$

Question 16

Suppose $$f(x) = x^4$ - 2x^2$ - 16x + 15. $$According to Descartes' Rule of Signs, how many negative real zeros are possible?

Accepted Answer

1

Question 17

A medical dosage model is given by $$f(x) = x^3$ - 2x^2$ - 5x + 6. If$$ $$x is $$the dosage in mg, which of the following is a valid positive rational zero?

Accepted Answer

$$3$$

Question 18

Given $$f(x) = x^3$ + 5x^2$ - x - 2$$ and $$x = -2 is a $$zero, what are the other zeros?

Accepted Answer

$$-1$$ and $$1$$

Question 19

If $$f(x) = x^3$ + 5x^2$ - x - 2$$ and you use synthetic division with $$x = 1$$, what is the remainder?

Accepted Answer

Question 20

Find a cubic polynomial with real coefficients, leading coefficient 1, and zeros at $$-5$$ and $$4 + 3i. $$What is the polynomial?

Accepted Answer

$$x^3$$ + $$5x^2 - 11x + 65$

Zeros of Polynomial Functions: Rational Zero Theorem and Descartes' Rule of Signs

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