Polynomials: Operations, Factoring, and Solving Equations

Question 1

Which of the following expressions is NOT a polynomial?

Accepted Answer

√x + 7

Answer

2x^2 + 3x - 5

Answer

x^3 - 2x + 1

Answer

(2x + 1)(x - 3)

Question 2

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

Accepted Answer

x = -2, -3

Answer

x = 2, 3

Answer

x = -3, 2

Answer

x = 3, -2

Question 3

What is the degree of the polynomial 3x^2 - 5x + 2?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 4

What is the remainder when the polynomial 2x^3 - 5x^2 + 3x - 1 is divided by (x - 2)?

Accepted Answer

-1

Answer

1

Answer

3

Answer

5

Question 5

What is the discriminant of the polynomial equation ax^2 + bx + c = 0?

Accepted Answer

b^2 - 4ac

Answer

4ac - b^2

Answer

b^2 + 4ac

Answer

-b^2 + 4ac

Question 6

What is the multiplicative inverse of the polynomial 2x - 1?

Accepted Answer

1/(2x - 1)

Answer

1/(2x + 1)

Answer

2/(x - 1)

Answer

2/(x + 1)

Question 7

Find the sum of the polynomials (x^2 - 2x + 3) and (2x^2 + x - 1).

Accepted Answer

3x^2 + x + 2

Answer

x^2 - x + 4

Answer

3x^2 - x + 2

Answer

x^2 + x + 2

Question 8

Sketch the graph of the polynomial f(x) = -x^2 + 4x - 3.

Accepted Answer

Parabola opening downwards with vertex at (2, -1)

Answer

Parabola opening upwards with vertex at (2, 1)

Answer

Parabola opening upwards with vertex at (-2, 1)

Answer

Parabola opening downwards with vertex at (-2, -1)

Question 9

Which of the following polynomials is irreducible over the field of real numbers?

Accepted Answer

x^2 + 1

Answer

x^2 - 2

Answer

x^2 - 4x + 3

Answer

x^2 + 2x + 1

Question 10

Find the least common multiple (LCM) of the polynomials 2x^2 - 4x and x^2 - 2x.

Accepted Answer

2x(x - 2)

Answer

x(2x - 4)

Answer

2x^2(x - 2)

Answer

x^2(2x - 4)

Question 11

Simplify the expression (2x - 1) / (x + 2).

Accepted Answer

2

Answer

-2

Answer

2x - 1

Answer

-2x + 1

Question 12

Identify the coefficient of the x^2 term in the polynomial 3x^4 - 2x^3 + 5x^2 - 7x + 1.

Accepted Answer

5

Answer

3

Answer

-2

Answer

-7

Question 13

Which of the following is NOT a valid operation for polynomials?

Accepted Answer

Differentiation

Answer

Addition

Answer

Subtraction

Answer

Multiplication

Question 14

Factor the polynomial: x^2 - 5x + 6.

Accepted Answer

(x - 2)(x - 3)

Answer

(x + 2)(x + 3)

Answer

(x - 6)(x + 1)

Answer

(x + 6)(x - 1)

Question 15

Solve the polynomial equation: 2x^2 - 5x + 3 = 0.

Accepted Answer

x = 1, x = 3

Answer

x = 1, x = 1.5

Answer

x = 1.5, x = 3

Answer

x = 2, x = 3

Question 16

Identify the statement that is NOT a characteristic of polynomials.

Accepted Answer

Polynomials can have complex coefficients.

Answer

Polynomials can be added, subtracted, multiplied, and divided.

Answer

Polynomials can have terms with any non-negative integer exponent.

Answer

The coefficients of a polynomial can be any real number.

Question 17

Find the degree of the polynomial **x^4 - 2x^3 + 5x^2 - 3x + 1**.

Accepted Answer

4

Answer

3

Answer

5

Answer

6

Question 18

Which of the following is a polynomial function?

Accepted Answer

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

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Answer

f(x) = log(x)

Question 19

Which of the following is a polynomial?

Accepted Answer

x^2 + 2x - 1

Answer

1 / x

Answer

sin(x)

Answer

e^x

Question 20

Simplify the expression: **(x + 2)(x - 3)**

Accepted Answer

x^2 - x - 6

Answer

x^2 + x - 6

Answer

x^2 - 5x - 6

Answer

x^2 + 5x - 6

Question 21

Which of the following is the degree of the polynomial **x^4 - 2x^3 + x^2 - 3x + 1**?

Accepted Answer

4

Answer

2

Answer

3

Answer

5

Question 22

Which of the following is a polynomial in the variable x?

Accepted Answer

2x³ + 5x - 1

Answer

1 / x

Answer

sin(x)

Answer

x^(1/2)

Question 23

Factorize the polynomial: x² - 4

Accepted Answer

(x + 2)(x - 2)

Answer

x(x - 4)

Answer

(x - 2)²

Answer

Cannot be factorized

Question 24

Which of the following expressions represents the degree of the polynomial x³ - 2x² + 5x - 7?

Accepted Answer

3

Answer

2

Answer

5

Answer

-7

Question 25

What is the degree of the polynomial 3x⁴ - 2x² + 5x - 1?

Accepted Answer

4

Answer

3

Answer

1

Answer

2

Question 26

Simplify the expression (2x + 3)(x - 1).

Accepted Answer

2x² + x - 3

Answer

2x² + 5x - 3

Answer

2x² - 3

Answer

2x² - x - 3

Question 27

Factor the polynomial x² - 9 completely.

Accepted Answer

(x + 3)(x - 3)

Answer

(x - 3)²

Answer

(x - 9)(x + 1)

Answer

(x + 3)²

Question 28

What is the remainder when x³ - 2x² + 5x - 1 is divided by x - 2?

Accepted Answer

11

Answer

5

Answer

-1

Answer

3

Question 29

Which of these is a factor of the polynomial x³ + 2x² - 5x - 6?

Accepted Answer

x + 3

Answer

x - 1

Answer

x - 2

Answer

x + 1

Question 30

Solve the equation x² - 4x + 3 = 0.

Accepted Answer

x = 1 or x = 3

Answer

x = -1 or x = -3

Answer

x = 1 or x = -3

Answer

x = 2 or x = -1

Question 31

What are the roots of the polynomial equation (x - 2)(x + 1) = 0?

Accepted Answer

x = 2 or x = -1

Answer

x = 2 or x = 1

Answer

x = -2 or x = -1

Answer

x = -2 or x = 1

Question 32

What is the maximum number of real roots the polynomial 2x³ - 5x² + 3x + 1 can have?

Accepted Answer

3

Answer

2

Answer

1

Answer

4

Question 33

Which of these describes a possible graph of the polynomial function f(x) = x³ - 2x² - x + 2?

Accepted Answer

A graph that rises to the right and has three x-intercepts.

Answer

A graph that falls to the right and has two x-intercepts.

Answer

A graph that falls to the right and has three x-intercepts.

Answer

A graph that rises to the right and has two x-intercepts.

Question 34

A rectangular garden has a length of (x + 5) meters and a width of (x - 2) meters. What is the area of the garden in terms of x?

Accepted Answer

x^2 + 3x - 10

Answer

x^2 - 3x - 10

Answer

x^2 - 10

Answer

x^2 + 7x - 10

Question 35

Which of the following is a polynomial?

Accepted Answer

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

Answer

1/x + 2

Answer

√(x)

Answer

sin(x)

Question 36

Simplify the polynomial (x + 2)(x - 3).

Accepted Answer

x^2 - x - 6

Answer

x^2 + 5x - 6

Answer

x^2 - 5x - 6

Answer

x^2 + x - 6

Question 37

Factor the polynomial x^2 - 4.

Accepted Answer

(x + 2)(x - 2)

Answer

(x - 2)^2

Answer

(x + 4)(x - 4)

Answer

Cannot be factored

Question 38

Solve the polynomial equation x^3 - 8 = 0.

Accepted Answer

x = 2

Answer

x = -2

Answer

x = 4

Question 39

Find the remainder when x^4 + 2x^2 - 1 is divided by x + 1.

Accepted Answer

-2

Answer

1

Answer

0

Answer

2

Question 40

Which of the following is a synthetic division of (x^3 - 2x + 1) ÷ (x - 1)?

Accepted Answer

1 x^2 + 1 x + 1

Answer

x^2 - x + 1

Answer

x^2 + x - 1

Answer

x^2 - 1 x + 1

Question 41

Find the degree of the polynomial 3x^5 - 2x^3 + x - 1.

Accepted Answer

5

Answer

4

Answer

6

Answer

3

Question 42

Which of the following is a monic polynomial?

Accepted Answer

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

Answer

2x^3 - 4x^2 + 2x - 2

Answer

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

Answer

3x^3 - 6x^2 + 3x - 3

Question 43

Which of the following is a reciprocal polynomial?

Accepted Answer

1 - x + x^2 - x^3

Answer

1 - x - x^2 + x^3

Answer

1 + x + x^2 + x^3

Answer

1 + x - x^2 - x^3

Question 44

Find the greatest common factor (GCF) of the polynomials 2x^3 - 4x^2 and 6x^2 - 12x.

Accepted Answer

2x

Answer

2

Answer

4x

Answer

x