Question 1

Which of the following correctly defines the imaginary unit $$i$$?

Accepted Answer

$$i = \sqrt{-1}$$

Question 2

What is the standard form of a complex number?

Accepted Answer

$$a + bi$$

Question 3

Which of the following is a purely imaginary number?

Accepted Answer

$$0 + 5i$$

Question 4

Evaluate $$\sqrt{-16}.$$

Accepted Answer

$$4i$$

Question 5

Which of the following is the correct way to multiply $$\sqrt{-4} \cdot \sqrt{-9}$$?

Accepted Answer

$$\sqrt{-4} \cdot \sqrt{-9} = 2i \cdot 3i = 6i^2$ = -6$$

Question 6

If $$a + bi$$ and $$c + di$$ are complex numbers, what is their product?

Accepted Answer

$$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$$

Question 7

What is the result of $$(3 + 2i)(3 - 2i)$$?

Accepted Answer

$$3^2$$ + $$(2)^2 = 9 + 4 = 13$

Question 8

Which of the following is the correct simplification of $$i^{17}$$?

Accepted Answer

$$i$$

Question 9

A quadratic function in standard form is given by:

Accepted Answer

$$P(x) = ax^2$ + bx + c$$

Question 10

Given the quadratic function $$f(x) = (x-3)^2$ - 8$$, what are the coordinates of the vertex?

Accepted Answer

$$(3, -8)$$

Question 11

What is the vertex of the graph of $$P(x) = ax^2$ + bx + c$$?

Accepted Answer

$$\left( \frac{-b}{2a}, P\left(\frac{-b}{2a}\right) \right)$$

Question 12

Which of the following is the center-radius form of the equation of a circle?

Accepted Answer

$$(x-h)^2$$ + $$(y-k)^2$$ = $$r^2$$

Question 13

Which of the following statements about the number of $$x-$$intercepts of a polynomial function of degree $$n is $$true?

Accepted Answer

The graph will have at most $$n$$ $$x-$$intercepts.

Question 14

If $$P(x) is $$divided by $$x-k$$, what does the Remainder Theorem state?

Accepted Answer

The remainder is $$P(k).$$

Question 15

Which of the following is the correct quadratic formula for solving $$ax^2 + bx + c = 0$?

Accepted Answer

$$x = \frac{-b \pm \sqrt{b^2$ - 4ac}}{2a}$$

Question 16

A real-world scenario: A company models its profit $$P(x) in $$thousands of dollars by the quadratic function $$P(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold in thousands. What is the maximum profit, and how many units should be sold to achieve it?

Accepted Answer

Maximum profit is $$26$$ thousand dollars at $$x = 2$$ thousand units.

Question 17

Which theorem guarantees that a polynomial $$P(x)$$ with real coefficients has at least one real zero between $$a$$ and $$b if$$ $$P(a)$$ and $$P(b)$$ have opposite signs?

Accepted Answer

Intermediate Value Theorem

Question 18

If $$P(x) = x^4$ - 3x^3$ - 5x^2$ + 2x - 16 is $$divided by $$x-3$$, what is the degree of the remainder?

Accepted Answer

Less than the degree of $$x-3$$, so degree $$0$$

Question 19

Which of the following is a correct application of the Zero-Product Property?

Accepted Answer

If $$(x-2)(x+3) = 0$$, then $$x = 2 or$$ $$x = -3$$

Complex Numbers, Quadratic and Polynomial Functions: Essential Precalculus Concepts

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