Question 1

Given the polynomial $$P(x) = 2x^4$ - 11x^3$ + 6x^2$ - 20x + 8$$, use Descartes' Rule of Signs to determine the possible number of positive real zeros.

Accepted Answer

$$2 or$$ $$0$$

Question 2

For the polynomial $$P(x) = x^4$ - 2x^3$ + 10x^2$ - 18x + 9$$, what does Descartes' Rule of Signs predict about the number of positive real zeros?

Accepted Answer

$$1 or$$ $$0$$

Question 3

Given $$P(x) = 6x^4$ - 41x^3$ + 89x^2$ - 56x - 12$$, use the Rational Root Theorem to list all possible rational zeros.

Accepted Answer

$$\pm1, \pm2, \pm3, \pm4, \pm6, \pm12, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{4}{3}, \pm\frac{6}{3}, \pm\frac{12}{3}$$

Question 4

A polynomial $$f(x)$$ with real coefficients has zeros at $$8$$, $$4-3i$$, and $$2i. $$What are all the zeros of $$f(x)$$?

Accepted Answer

$$8, 4-3i, 4+3i, 2i, -2i$$

Question 5

Given $$P(x) = x^3$ - 5x^2$ + 7x - 3$$, use the Rational Root Theorem to list all possible rational zeros.

Accepted Answer

$$\pm1, \pm3$$

Question 6

Suppose $$P(x) is a $$degree 4 polynomial with real coefficients and zeros at $$1 + 3i$$ and $$-1. $$Which of the following must also be a zero of $$P(x)$$?

Accepted Answer

$$1 - 3i$$

Question 7

If $$P(x) = x^4$ - 2x^3$ + 10x^2$ - 18x + 9$$, what are all its real zeros?

Accepted Answer

None; all zeros are nonreal complex numbers.

Question 8

Given $$P(x) = 3x^4$ - 17x^3$ + 5x^2$ - 29x + 20$$, use synthetic division to test if $$x = 5 is a $$zero.

Accepted Answer

Yes, $$x=5 is a $$zero.

Question 9

A degree 5 polynomial with real coefficients has zeros at $$2$$, $$-1 ($$with multiplicity 2), and $$3+2i. $$Write the polynomial in factored form.

Accepted Answer

$$(x-2)(x+1)^2(x-(3+2i))(x-(3-2i$$))$$

Question 10

Given $$P(x) = x^3$ - 5x^2$ + 7x - 3$$, use synthetic division to determine the remainder when divided by $$x-1.$$

Accepted Answer

$$0$$

Question 11

If $$P(x) is a $$polynomial with integer coefficients and $$\frac{2}{3} is a $$rational zero, which of the following must divide the leading coefficient?

Accepted Answer

$$3$$

Question 12

Given $$P(x) = x^4$ - 2x^3$ + 10x^2$ - 18x + 9$$, which of the following is a possible rational zero according to the Rational Root Theorem?

Accepted Answer

$$1$$

Question 13

A polynomial $$P(x)$$ with real coefficients has $$3+2i as a $$zero. Which of the following must also be a zero?

Accepted Answer

$$3-2i$$

Question 14

Given $$P(x) = x^3$ - 5x^2$ + 7x - 3$$, which of the following is NOT a possible rational zero?

Accepted Answer

$$2$$

Question 15

If $$P(x) is a $$degree 4 polynomial with real coefficients and zeros at $$1+\sqrt{3}$$ and $$-2$$, what are all the zeros?

Accepted Answer

$$1+\sqrt{3}, 1-\sqrt{3}, -2, -2$$

Question 16

Given $$P(x) = x^4$ - 2x^3$ + 10x^2$ - 18x + 9$$, use the quadratic formula to find the zeros of $$x^2 - 2x + 10 = 0$.

Accepted Answer

$$x = 1 \pm 3i$$

Question 17

If $$P(x) is a $$degree 4 polynomial with real coefficients and zeros at $$2$$, $$-4$$, and $$1+\sqrt{3}$$, write $$P(x) in $$factored form.

Accepted Answer

$$(x-2)(x+4)(x-(1+\sqrt{3}))(x-(1-\sqrt{3}))$$

Question 18

Given $$P(x) = x^3$ - 5x^2$ + 7x - 3$$, if $$x=1 is a $$zero, what is the depressed polynomial after dividing by $$x-1$$?

Accepted Answer

$$x^2 - 4x + 3$

Question 19

A degree 4 polynomial with real coefficients has zeros at $$1$$, $$-2$$, and $$2i. $$What is the complete set of zeros?

Accepted Answer

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

Polynomial Zeros: Descartes' Rule of Signs, Rational Root Theorem, and Complex Conjugates

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