Algebra II Online Quizzes: Master Advanced Algebraic Concepts

Question 1

When solving a quadratic equation using the quadratic formula, what do complex number solutions imply?

Accepted Answer

The equation has no real solutions.

Answer

The equation has two additional real solutions.

Answer

The equation is only solvable if its coefficients are complex numbers.

Answer

Complex solutions are irrelevant and can be disregarded.

Question 2

For a quadratic equation in the form ax^2+bx+c=0, what does the discriminant tell us about the equation's solutions?

Accepted Answer

Positive discriminant: Two distinct real solutions

Accepted Answer

Zero discriminant: One real solution

Accepted Answer

Negative discriminant: No real solutions

Question 3

Identify the domain of the rational expression (x - 3)/(x^2 - 9).

Accepted Answer

{x : x ≠ ±3}

Answer

{x : x ≠ 0}

Answer

{x : all real numbers}

Answer

{x : x ≠ 9}

Question 4

Determine the degree of the polynomial function f(x) = 4x^5 - 3x^3 + 2x^2 - 5.

Accepted Answer

5

Answer

2

Answer

3

Answer

4

Question 5

Simplify the rational expression (x^2 - 4)/(x - 2) by factoring and canceling.

Accepted Answer

x - 2

Answer

x + 2

Answer

x^2 + 2

Answer

x^2 - 2

Question 6

A circle is inscribed in a square. If the area of the circle is 100π square units, what is the area of the square?

Accepted Answer

400

Answer

200

Answer

600

Answer

800

Question 7

Factor the polynomial expression: x³ - 3x² + 2x - 6

Accepted Answer

(x - 3)(x² + 2x + 2)

Answer

(x + 3)(x² - 2x + 2)

Answer

(x - 2)(x² + 3x + 3)

Answer

(x + 2)(x² - 3x + 3)

Question 8

Simplify the rational expression: (x² - 9) / (x - 3)

Accepted Answer

x + 3

Answer

x - 3

Answer

x² + 3

Answer

x² - 3

Question 9

Which graph represents the function f(x) = -x² + 4x + 3?

Accepted Answer

A downward-opening parabola with a vertex at (2, 7)

Answer

An upward-opening parabola with a vertex at (-2, 7)

Answer

A straight line with a slope of -1 and a y-intercept of 3

Answer

A circle with a radius of 2 and a center at (2, 7)

Question 10

Which of the following indicates the degree of the polynomial function f(x) = 3x^4 - 2x^3 + 5x - 1?

Accepted Answer

4

Answer

2

Answer

3

Answer

5

Question 11

Factor the polynomial expression x^3 - 8:

Accepted Answer

(x - 2)(x^2 + 2x + 4)

Answer

(x + 2)(x^2 - 2x + 4)

Answer

(x - 2)^3

Answer

(x + 2)^3

Question 12

Find the domain of the function f(x) = sqrt(x - 2):

Accepted Answer

[2, ∞)

Answer

[-2, ∞)

Answer

(-∞, 2]

Answer

(2, ∞)

Question 13

Identify the characteristics of the graph of the polynomial function f(x) = x^2 - 2x + 1:

Accepted Answer

A parabola opening upwards with its vertex at (1, 0)

Answer

A constant function with a value of 1

Answer

A parabola opening downwards with its vertex at (1, 0)

Answer

A linear function with a slope of 1

Question 14

Simplify the expression: (2x^2y)^-3

Accepted Answer

1/8x^6y^-3

Answer

6x^3y^6

Answer

8x^3y^-6

Answer

1/6x^6y^3

Question 15

Identify the domain of the function f(x) = sqrt(x + 2):

Accepted Answer

x ≥ -2

Answer

x ≤ -2

Answer

All real numbers

Answer

x > -2

Question 16

Which of the following expressions represents a polynomial function?

Accepted Answer

f(x) = 3x^2 - 2x + 5

Answer

h(x) = √(x + 4)

Answer

j(x) = 1/x - 2

Answer

g(x) = 2^x + 1

Question 17

What is the degree of the polynomial  p(x) = 7x^4 - 3x^2 + 2x - 1?

Accepted Answer

4

Answer

3

Answer

2

Answer

1

Question 18

Simplify the rational expression: (x^2 - 4) / (x + 2)

Accepted Answer

x - 2

Answer

(x - 2)^2

Answer

x + 2

Answer

x^2 - 2

Question 19

What is the vertex of the parabola defined by the function g(x) = (x - 2)^2 + 1?

Accepted Answer

(2, 1)

Answer

(2, -1)

Answer

(-2, 1)

Answer

(-2, -1)

Question 20

The function h(x) = -2(x + 1)^2 + 3 represents a parabola that opens in which direction?

Accepted Answer

Downwards

Answer

To the right

Answer

Upwards

Answer

To the left

Question 21

What is the domain of the function j(x) = √(x - 4)?

Accepted Answer

x ≥ 4

Answer

x ≤ 4

Answer

x ≠ 4

Answer

All real numbers

Question 22

What is the degree of the polynomial function  *f(x) = 3x⁴ - 2x³ + x² - 5*?

Accepted Answer

4

Answer

2

Answer

5

Answer

3

Question 23

What is the remainder when the polynomial *f(x) = x³ - 3x² + 2x - 1* is divided by *(x - 1)*?

Accepted Answer

-1

Answer

2

Answer

0

Answer

1

Question 24

What is the equation of the horizontal asymptote of the rational function *f(x) = (x - 2) / (x² - 4)*?

Accepted Answer

y = 0

Answer

y = 2

Answer

y = 1

Answer

y = -1

Question 25

Solve for *x*:  2(x - 1) + 3(x + 2) = 5(x - 3)

Accepted Answer

x = 10

Answer

x = 20

Answer

x = 15

Answer

x = 5

Question 26

How many real solutions does the equation *x⁴ - 8x² + 16 = 0* have?

Accepted Answer

2

Answer

1

Answer

4

Answer

0

Question 27

Which graph represents the polynomial function *f(x) = x² - 4x + 3*?

Accepted Answer

A parabola opening upward

Answer

A line

Answer

A circle

Answer

A parabola opening downward

Question 28

Solve the following system of equations:
2x + y = 7
3x - y = 1

Accepted Answer

x = 2, y = 3

Answer

x = -2, y = -3

Answer

x = 3, y = 2

Answer

x = -3, y = -2

Question 29

What is the discriminant of the quadratic equation *2x² - 5x + 3 = 0*?

Accepted Answer

-7

Answer

11

Answer

-11

Answer

7

Question 30

Factor the polynomial *x³ - 2x² - 5x + 6* completely.

Accepted Answer

(x - 1)(x - 2)(x + 3)

Answer

(x - 1)(x + 2)(x - 3)

Answer

(x + 1)(x - 2)(x + 3)

Answer

(x - 1)(x - 2)(x - 3)

Question 31

What is the inverse function of *f(x) = 2x + 3*?

Accepted Answer

f⁻¹(x) = (x - 3) / 2

Answer

f⁻¹(x) = 2x - 3

Answer

f⁻¹(x) = (x + 3) / 2

Question 32

What is the degree of the polynomial function f(x) = 3x^4 - 2x^2 + 5x - 1?

Accepted Answer

4

Answer

2

Answer

3

Answer

1

Question 33

Which of the following is a factor of the polynomial x^3 - 2x^2 - 5x + 6?

Accepted Answer

(x - 3)

Answer

(x - 2)

Answer

(x + 1)

Answer

(x + 2)

Question 34

What is the horizontal asymptote of the rational function f(x) = (2x^2 + 1) / (x^2 - 4)?

Accepted Answer

y = 2

Answer

x = -2

Answer

y = 0

Answer

x = 2

Question 35

The graph of a polynomial function has a zero at x = -2 with multiplicity 3. What does this tell us about the behavior of the graph at x = -2?

Accepted Answer

The graph touches the x-axis at x = -2 and changes direction.

Answer

The graph has a vertical asymptote at x = -2.

Answer

The graph crosses the x-axis at x = -2.

Question 36

Solve the system of equations: 2x + 3y = 7 and x - 2y = 1.

Accepted Answer

x = 2, y = 1

Answer

x = 3, y = -1

Answer

x = 1, y = 2

Answer

x = -1, y = 3

Question 37

What is the inverse function of f(x) = 2x + 3?

Accepted Answer

f^-1(x) = (x - 3) / 2

Answer

f^-1(x) = -2x - 3

Answer

f^-1(x) = (x + 3) / 2

Answer

f^-1(x) = 2x - 3

Question 38

Simplify the rational expression: (x^2 - 9) / (x^2 + 6x + 9).

Accepted Answer

(x - 3) / (x + 3)

Answer

(x + 3) / (x - 3)

Answer

x + 3

Answer

x - 3

Question 39

What is the solution to the equation |2x - 1| = 5?

Accepted Answer

x = 3 or x = -2

Answer

x = 3

Answer

x = -2

Answer

x = 2 or x = -3

Question 40

Find the vertex of the parabola defined by the equation y = -2x^2 + 4x - 1.

Accepted Answer

(1, 1)

Answer

(0, -1)

Answer

(-1, 7)

Answer

(2, -1)

Question 41

The graph of a function f(x) is shown. What is the domain of the function?

Accepted Answer

[-3, 4]

Answer

(-2, 3)

Answer

(-3, 4)

Answer

[-2, 3]