Polynomials and Rational Expressions: Online Quizzes and Practice

Question 1

Find the Least Common Multiple (LCM) of the fractions 1/3 and 2/5:

Accepted Answer

15

Answer

6

Answer

10

Answer

30

Question 2

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

Accepted Answer

9x^2 + 12xy + 4y^2

Answer

9x^2 - 12xy + 4y^2

Answer

9x^2 + 4y^2

Answer

3x^2 + 2y^2

Question 3

Solve the equation: 2x(x - 3) = 0

Accepted Answer

x = 0, x = 3

Answer

x = 2, x = -3

Answer

x = 0

Answer

x = 3

Question 4

Simplify the rational expression: (x² - 4) / (x + 2)

Accepted Answer

x - 2

Answer

x + 2

Answer

(x - 2)(x + 2)

Answer

x - 4

Question 5

Determine the horizontal asymptote of the rational function: f(x) = (2x² - 1) / (x² + 1)

Accepted Answer

y = 1

Answer

y = 2

Answer

y = -1

Answer

No horizontal asymptote

Question 6

Expand and simplify the expression: (2x + 3)(x - 5)

Accepted Answer

2x^2 - 7x - 15

Answer

2x^2 + 7x + 15

Answer

2x^2 - 7x + 15

Answer

2x^2 + 7x - 15

Question 7

Which of the following expresiones represents a polynomial?

Accepted Answer

x^2 - 2x + 1

Answer

√(x + 2)

Answer

1 / (x - 3)

Answer

sin(2x)

Question 8

Which is the coefficient of the term 3x^2 in the polynomial 2x^3 - 3x^2 + 4x - 1?

Accepted Answer

-3

Answer

0

Answer

2

Answer

4

Question 9

Factor the polynomial x^2 - 4.

Accepted Answer

(x - 2)(x + 2)

Answer

(x - 2)^2

Answer

(x + 2)^2

Answer

Cannot be factored

Question 10

Simplify the fraction (x^2 - 4) / (x - 2).

Accepted Answer

x + 2

Answer

x - 2

Answer

x^2 + 2x + 4

Answer

Cannot be simplified

Question 11

Which of the following is a rational function?

Accepted Answer

1 / (x - 2)

Answer

x^2 + 2x - 3

Answer

√(x)

Answer

e^x

Question 12

Which characteristic is NOT true for all polynomial functions?

Accepted Answer

They can have discontinuities.

Answer

They can have complex roots.

Answer

They can have any degree.

Answer

They must have at least one x-intercept.

Question 13

Which of the following is NOT a fundamental property of polynomials?

Accepted Answer

Roots can always be found using the Quadratic Formula

Answer

Closed under addition, subtraction, and multiplication

Answer

Degree can only be a non-negative integer

Answer

Coefficients can be any real number

Question 14

Factor the cubic polynomial x^3 + 2x^2 - 5x - 6 into linear factors.

Accepted Answer

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

Answer

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

Answer

(x - 1)(x^2 + 1)

Answer

The polynomial cannot be factored.

Question 15

Which of the following is NOT a term of the polynomial 3x^2 - 5x + 2?

Accepted Answer

x

Answer

3x^2

Answer

-5x

Answer

2

Question 16

Factor the polynomial: x^2 - 9

Accepted Answer

(x + 3)(x - 3)

Answer

(x - 3)(x - 3)

Answer

(x + 9)(x - 9)

Answer

Cannot be factored

Question 17

Which of the following expressions is a polynomial?

Accepted Answer

3x^2 - 2x + 1

Answer

x^(1/2) + 2x

Answer

(x+1)/(x-1)

Answer

sin(x) + cos(x)

Question 18

Simplify the polynomial expression: (2x^3 - 3x^2 + 4) - (x^3 + 2x^2 - 1)

Accepted Answer

x^3 - 5x^2 + 5

Answer

3x^3 - x^2 + 3

Answer

x^2 - x^3 + 4

Answer

x^3 - x^2 - 3

Question 19

Solve the polynomial equation: x^2 - 5x + 6 = 0

Accepted Answer

x = 2, x = 3

Answer

x = -2, x = -3

Answer

x = 1, x = 6

Answer

The equation has no real solutions

Question 20

Which of the following algebraic expressions is a polynomial?

Accepted Answer

x^2 + 2x - 3

Answer

x^2 - 2x + 3 / x + 1

Answer

1 / x^2

Answer

sin(x)

Question 21

Factor the polynomial x^2 - 9.

Accepted Answer

(x - 3)(x + 3)

Answer

(x + 3)^2

Answer

(x - 3)^2

Answer

Cannot be factored

Question 22

Which of the following is NOT a property of rational expressions?

Accepted Answer

A rational expression can be equal to zero if the numerator is zero and the denominator is not.

Answer

The product of two rational expressions is a rational expression.

Answer

The sum of two rational expressions with the same denominator is a rational expression.

Answer

The quotient of two rational expressions is a rational expression.

Question 23

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

Accepted Answer

x ≥ 2

Answer

x < 2

Answer

x > 2

Answer

All real numbers

Question 24

Which of the following functions is a rational function?

Accepted Answer

x / (x + 1)

Answer

x^2 + 1

Answer

sin(x)

Answer

e^x

Question 25

Which of the following is NOT a polynomial?

Accepted Answer

1 / x

Answer

x² + 2x - 3

Answer

2x³ - x + 5

Answer

-x⁵ + 3x² - 1

Question 26

Factor the polynomial expression: x² - 9

Accepted Answer

(x + 3)(x - 3)

Answer

(x - 3)²

Answer

Cannot be factored

Answer

(x + 3)³

Question 27

What is the least common denominator (LCD) of the fractions 1/x and 1 / (x + 1)?

Accepted Answer

x(x + 1)

Answer

x + 1

Answer

x² + x

Answer

x

Question 28

Solve the inequality: x² - 4x + 3 > 0

Accepted Answer

x < 1 or x > 3

Answer

1 < x < 3

Answer

x < 3 or x > 1

Answer

-1 < x < 3

Question 29

What is the domain of the rational expression 1 / (x - 2)?

Accepted Answer

x ≠ 2

Answer

All real numbers

Answer

x ≥ 0

Answer

x ≠ 0

Question 30

Determine the domain of the function: f(x) = √(x - 2)

Accepted Answer

[2, ∞)

Answer

[2, ∞]

Answer

(-∞, 2)

Answer

(-∞, 2]

Question 31

Find the remainder when dividing the polynomial x^3 - 2x + 1 by x + 1 using synthetic division or long division

Accepted Answer

-2

Answer

1

Answer

0

Question 32

Which of the following is NOT a term of the polynomial 3x^3 + 2x^2 - 5x + 1?

Accepted Answer

x^4

Answer

1

Answer

3x^3

Answer

-5x

Question 33

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

Accepted Answer

x ≥ 2

Answer

x ≤ 2

Answer

All real numbers

Question 34

Which of the following expressions is a polynomial?

Accepted Answer

(2x^3) + (5x^2) - x - 1

Answer

(x^2 + 2) / (x - 1)

Answer

(x^(1/2)) + 3

Question 35

Which expression represents the factored form of the polynomial x² - 9?

Accepted Answer

(x - 3)(x + 3)

Answer

(x - 9)(x + 1)

Answer

(x - 3)²

Answer

(x + 9)(x - 1)

Question 36

What is the product of the rational expressions (x + 1)/(x - 2) and (x - 2)/(x + 3)?

Accepted Answer

(x + 1)/(x + 3)

Answer

x + 1

Answer

1

Answer

x - 3

Question 37

Which of the following is NOT a polynomial expression?

Accepted Answer

x^-2

Answer

3x^4 - 2x^2 + 7

Answer

2x^3 + 5x^2 - 1

Answer

(x-2)^2

Question 38

Simplify the following expression: (2x^2 - 3x + 1) + (x^2 + 2x - 5)

Accepted Answer

3x^2 - x - 4

Answer

3x^2 + x - 4

Answer

3x^2 + 5x - 6

Answer

x^2 - x - 4

Question 39

Factor the following polynomial: x^2 - 4

Accepted Answer

(x+2)(x-2)

Answer

(x-2)(x+4)

Answer

(x+2)^2

Answer

(x+1)(x-4)

Question 40

Solve the following equation for x: 2x^2 - 5x + 2 = 0

Accepted Answer

x=1/2 or x=2

Answer

x=1 or x=4

Answer

No real solutions

Answer

x=-1 or x=-2

Question 41

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

Accepted Answer

x-2

Answer

Cannot be simplified

Answer

(x-2)^2

Answer

x+2

Question 42

Which of the following is equivalent to the rational expression (x+2)/(x-3)?

Accepted Answer

1 + 5/(x-3)

Answer

5/(x-3) - 1

Answer

1 - 5/(x-3)

Question 43

Which of the following is the domain of the function f(x) = (x-1)/(x^2+2x-3)?

Accepted Answer

All real numbers except x=-1 and x=3

Answer

Only x=-1 and x=3

Answer

All real numbers except x=1 and x=3

Answer

All real numbers