Polynomials: Operations, Factoring, and Applications

Question 1

Which expression is a polynomial?

Accepted Answer

x² + 2x + 1

Answer

√x + 3

Answer

sin(x) + 1

Answer

e^x + 1

Question 2

Find the degree of the polynomial: 3x⁴ - 2x² + 5

Accepted Answer

4

Answer

2

Answer

5

Answer

6

Question 3

Factor the polynomial: x² - 4

Accepted Answer

(x - 2)(x + 2)

Answer

(x - 4)(x + 4)

Answer

(x - 2)

Answer

Cannot be factored

Question 4

In polynomial division, what does the remainder signify?

Accepted Answer

The remainder is the part of the dividend that cannot be divided evenly by the divisor.

Answer

The remainder is always zero.

Answer

The remainder is the same as the quotient of the division.

Answer

The remainder is not important in polynomial division.

Question 5

Multiply the polynomials f(x) = 2x - 1 and g(x) = x² + 3x - 4.

Accepted Answer

2x³ + 5x² - 10x + 4

Answer

2x³ - 5x² - 10x + 4

Answer

x³ - 5x² - 10x + 4

Answer

x³ + 5x² - 10x + 4

Question 6

Find the degree of the polynomial f(x) = 5x⁴ - 2x² + 3x - 1.

Accepted Answer

4

Answer

3

Answer

1

Answer

2

Question 7

Which of the following is a polynomial function?

Accepted Answer

f(x) = 2x³ - 5x² + 7x - 1

Answer

k(x) = e^x

Answer

g(x) = 1/x

Answer

h(x) = sin(x)

Question 8

Identify the expression that represents a polynomial in one variable.

Accepted Answer

x^2 + 3x - 5

Answer

2x^2y + 5xy

Answer

e^(2x)

Answer

sin(x) + cos(x)

Question 9

Factor the quadratic polynomial: x^2 - 9

Accepted Answer

(x - 3)(x + 3)

Answer

(x + 3)^2

Answer

(x - 3)^2

Answer

(x - 9)(x + 1)

Question 10

Which of the following is a rational polynomial?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 11

Identify the degree of the polynomial: 3x^4 - 5x^2 + 2x - 1

Accepted Answer

4

Answer

3

Answer

5

Answer

2

Question 12

Determine the constant term of the polynomial: 2x^3 - 5x^2 + 7x - 3

Accepted Answer

-3

Answer

7

Answer

2

Answer

0

Question 13

Sketch the graph of the polynomial: y = x^2 - 4x + 3

Accepted Answer

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

Answer

Line with slope -4 and y-intercept 3

Answer

Circle centered at (2, -1) with radius 3

Answer

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

Question 14

Factor the polynomial: x² - 4

Accepted Answer

(x + 2)(x - 2)

Answer

(x + 2)²

Answer

(x - 2)²

Answer

(x + 4)(x - 4)

Question 15

Factor the polynomial: x³ - 8

Accepted Answer

(x - 2)(x² + 2x + 4)

Answer

(x + 2)³

Answer

(x - 2)³

Answer

(x + 2)(x² - 2x + 4)

Question 16

Determine whether the polynomial: x³ - 2x² + x - 2 is divisible by x - 2

Accepted Answer

Yes

Answer

Cannot be determined

Answer

No

Answer

Only when x = 2

Question 17

Which of the following is not a characteristic of a polynomial?

Accepted Answer

It contains trigonometric functions.

Answer

It can be simplified using the rules of exponents and algebra.

Answer

It is an expression consisting of variables and coefficients.

Question 18

Divide the polynomial (x^3 - 3x^2 + 2x - 1) by the polynomial (x - 1)

Accepted Answer

x^2 - 2x + 1

Answer

x^2 + 2x - 1

Answer

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

Answer

x - 1

Question 19

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

Accepted Answer

6xy^3

Answer

12x^3y^7

Answer

2x^2y^4

Answer

3x^2y^3

Question 20

Which of the following expressions is NOT a polynomial?

Accepted Answer

x^2 + sin(x)

Answer

x^3 - 2x + 5

Answer

2x^4 - 3x^2 + x - 1

Answer

5x^2 - 7x + 3

Question 21

Determine the remainder when x^3 - 2x^2 + x - 2 is divided by x - 1

Accepted Answer

-3

Answer

1

Answer

-2

Answer

2

Question 22

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

Accepted Answer

4

Answer

2

Answer

3

Answer

1

Question 23

Which of the following expressions represents a monomial?

Accepted Answer

5x³

Answer

2x² - 3x + 4

Answer

x² + 2x - 1

Answer

x³ - 1

Question 24

Which of the following correctly describes the polynomial x⁴ - 5x² + 4?

Accepted Answer

It is a quartic polynomial.

Answer

It is a quadratic polynomial.

Answer

It is a linear polynomial.

Answer

It is a cubic polynomial.

Question 25

Simplify the expression (3x² + 2x - 1) + (x² - 4x + 5).

Accepted Answer

4x² - 2x + 4

Answer

2x² - 2x + 4

Answer

4x² + 6x - 6

Question 26

Factor the expression x² - 9.

Accepted Answer

(x + 3)(x - 3)

Answer

(x - 9)(x + 1)

Answer

(x + 9)(x - 1)

Answer

(x - 3)(x - 3)

Question 27

Which of the following expressions is a polynomial?

Accepted Answer

3x^2 + 2x - 1

Answer

1/x + 3x + 2

Answer

2x^-2 + x + 7

Answer

x^1/2 + 4x - 5

Question 28

Factor the polynomial x^2 - 9

Accepted Answer

(x + 3)(x - 3)

Answer

(x - 3)(x - 3)

Answer

(x + 3)(x + 3)

Answer

(x + 9)(x - 1)

Question 29

What is the degree of the polynomial 3x^4 - 2x^2 + 5x - 1?

Accepted Answer

4

Answer

2

Answer

3

Answer

1

Question 30

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

Accepted Answer

3x^2 - 2x + 1

Answer

3x^2 + 8x + 1

Answer

3x^2 - 8x + 1

Answer

x^2 - 2x + 1

Question 31

Multiply the following binomials: (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 32

Divide the polynomial 2x^3 + 5x^2 - 4x + 1 by x + 2. What is the remainder?

Accepted Answer

9

Answer

-9

Answer

-1

Answer

1

Question 33

Which of the following is a factor of the polynomial x^2 - 4?

Accepted Answer

(x + 2)

Answer

(x - 1)

Answer

(x + 1)

Answer

(x - 4)

Question 34

Factor the following polynomial completely: 2x^3 - 8x

Accepted Answer

2x(x + 2)(x - 2)

Answer

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

Answer

2(x^3 - 4x)

Answer

2x(x^2 - 4)

Question 35

Which of the following is a polynomial with a leading coefficient of 2 and a constant term of -5?

Accepted Answer

2x^3 - 3x^2 + 4x - 5

Answer

x^3 - 3x^2 + 4x - 5

Answer

2x^3 + 3x^2 + 4x - 5

Answer

2x^3 - 3x^2 + 4x + 5

Question 36

What is the simplified form of (x + y)^2?

Accepted Answer

x^2 + 2xy + y^2

Answer

x^2 + y^2

Answer

x^2 - 2xy + y^2

Answer

x^2 - y^2

Question 37

A rectangular garden has a length of (2x + 5) meters and a width of (x - 3) meters. What is the area of the garden in terms of x?

Accepted Answer

2x^2 - x - 15

Answer

2x^2 + 15

Answer

2x^2 - 15

Answer

2x^2 + x - 15

Question 38

The sum of two polynomials is 3x^3 + 2x^2 - 5x + 1. If one of the polynomials is x^3 - 4x^2 + 3x - 2, what is the other polynomial?

Accepted Answer

2x^3 + 6x^2 - 8x + 3

Answer

2x^3 - 6x^2 + 8x - 3

Answer

4x^3 + 2x^2 + 2x + 1

Answer

4x^3 - 2x^2 - 2x - 1

Question 39

Which of the following is NOT a polynomial?

Accepted Answer

√x + 2

Answer

x³ + 4x² - 2x

Answer

2x⁴ - 7

Answer

3x² - 5x + 1

Question 40

What is the degree of the polynomial 5x³ - 2x² + 7x - 1?

Accepted Answer

3

Answer

2

Answer

4

Answer

1

Question 41

Simplify the expression (3x² + 2x - 1) + (x² - 4x + 5).

Accepted Answer

4x² - 2x + 4

Answer

4x² + 6x + 4

Answer

2x² - 2x + 4

Answer

2x² + 6x + 4

Question 42

Subtract the polynomial (2x³ - 5x + 1) from (x³ + 3x² - 2x).

Accepted Answer

-x³ + 3x² + 3x - 1

Answer

3x³ + 3x² - 7x + 1

Answer

-x³ - 3x² + 3x - 1

Answer

-x³ + 3x² - 7x + 1

Question 43

Expand the product (x + 2)(x - 3).

Accepted Answer

x² - x - 6

Answer

x² + x - 6

Answer

x² + 5x - 6

Answer

x² - 5x - 6

Question 44

What is the quotient when 6x³ + 5x² - 2x + 1 is divided by 2x + 1?

Accepted Answer

3x² + x - 3/2

Answer

3x² - x - 1/2

Answer

3x² + x - 1/2

Answer

3x² - x + 3/2

Question 45

Factor the polynomial x² - 9.

Accepted Answer

(x + 3)(x - 3)

Answer

(x - 3)²

Answer

(x + 9)(x - 9)

Answer

(x + 3)²

Question 46

Factor the polynomial 2x² + 5x + 3.

Accepted Answer

(2x + 3)(x + 1)

Answer

(2x - 1)(x - 3)

Answer

(2x + 1)(x + 3)

Answer

(2x - 3)(x - 1)

Question 47

Which of the following is a factor of the polynomial x³ - 8?

Accepted Answer

x - 2

Answer

x² + 2

Answer

x + 2

Answer

x² - 2

Question 48

Solve the equation x² - 5x + 6 = 0 by factoring.

Accepted Answer

x = 2 or x = 3

Answer

x = -1 or x = -6

Answer

x = 1 or x = 6

Answer

x = -2 or x = -3