Polynomials: Definition, Operations, and Applications in Nursing

Question 1

Which of the following statements about polynomials is incorrect?

Accepted Answer

Polynomials cannot have complex coefficients.

Answer

Polynomials are closed under addition, subtraction, and multiplication.

Answer

The degree of a polynomial is the highest exponent of the variable.

Question 2

Multiply the polynomial (3x - 2) by the polynomial (2x^2 + 5x - 1).

Accepted Answer

6x^3 + 13x^2 - 7x + 2

Answer

6x^3 - 13x^2 + 7x - 2

Answer

6x^3 - 13x^2 - 7x + 2

Answer

6x^3 + 13x^2 + 7x - 2

Question 3

Divide the polynomial (x^4 - 2x^3 + 5x^2 - 7x + 3) by the polynomial (x - 3) using long division.

Accepted Answer

x^3 + x^2 - x + 1

Answer

x^3 - x^2 + x - 1

Answer

x^3 + x^2 + x + 1

Answer

x^3 - x^2 - x - 1

Question 4

Factor the polynomial x^2 - 4x + 4.

Accepted Answer

(x - 2)^2

Answer

(x + 2)^2

Answer

x(x - 4)

Answer

It cannot be factored

Question 5

Identify the degree of the polynomial 5x^3 - 2x^2 + 7x - 3.

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 6

Determine the leading coefficient of the polynomial 6x^4 - 3x^2 + 5x - 1.

Accepted Answer

6

Answer

1

Answer

3

Answer

5

Question 7

Evaluate the polynomial (x^2 - 2x + 3) at x = 2.

Accepted Answer

5

Answer

1

Answer

3

Answer

7

Question 8

Identify the monomial among the following options:

Accepted Answer

4x

Answer

2x^2 + 3x - 1

Answer

(x + 1)(x - 1)

Answer

x^2 - 2x + 1

Question 9

Which of the following is a binomial?

Accepted Answer

x + 2

Answer

3x^2

Answer

(x - 1)(x + 1)

Answer

x^3 - 2x^2 + x

Question 10

Identify the trinomial among the following options:

Accepted Answer

x^2 + 2x - 1

Answer

2x

Answer

(x + 1)

Answer

x^3 - x^2 + x

Question 11

Determine the degree of the polynomial P(x) = 3x^4 - 2x^2 + 5:

Accepted Answer

4

Answer

2

Answer

3

Answer

5

Question 12

For the polynomial P(x) = (x - 2)(x + 3), which of the following statements is true?

Accepted Answer

P(x) has two real zeros.

Answer

P(x) has no real zeros.

Answer

P(x) has one real zero.

Answer

Cannot be determined.

Question 13

Find the sum of the polynomials (2x^2 + 3x - 5) and (x^2 - 2x + 1).

Accepted Answer

3x^2 + 5x - 4

Answer

3x^2 + x - 6

Answer

3x^2 - x - 4

Answer

3x^2 - 5x - 6

Question 14

Which of the following polynomials has a degree of 4?

Accepted Answer

2x^4 - 5x^3 + 3x - 1

Answer

2x^2 + 3x - 5

Answer

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

Answer

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

Question 15

Which of the following is the graph of the polynomial f(x) = x^2 - 2x + 1?

Accepted Answer

A parabola opening upward

Answer

A parabola opening downward

Answer

A circle

Answer

An ellipse

Question 16

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

Accepted Answer

x^2 - x + 1

Answer

x^2 + x - 1

Answer

x^3 - x^2 + x

Answer

x^3 - 2x^2 + x

Question 17

Identify a factor of the polynomial: x^3 - 8

Accepted Answer

x - 2

Answer

x + 2

Answer

x^2 + 2x + 4

Answer

x^2 - 2x + 4

Question 18

Identify the monomial expression:

Accepted Answer

2x^3

Answer

x^2 + 2x + 1

Answer

x^2 * y^3

Answer

x * (2x - 1)

Question 19

Identify the binomial expression:

Accepted Answer

x^2 + y^2

Answer

x^3 - 2x + 1

Answer

x^3 * y^2

Answer

x * (2x - 1)

Question 20

Simplify the following polynomial expression step-by-step, showing your work:
(4x^3 - 3x^2 + 2x + 1) - (2x^3 + x^2 - 3x + 5)

Accepted Answer

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

Answer

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

Answer

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

Answer

2x^3 - 4x^2 - 5x + 6

Question 21

Which of the following mathematical expressions represents a polynomial?

Accepted Answer

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

Answer

1/x

Answer

(x + 1)/(x - 2)

Answer

sin(x) + cos(x)

Question 22

Find the degree of the polynomial P(x) = 4x^4 - 5x^2 + 3.

Accepted Answer

4

Answer

2

Answer

3

Answer

5

Question 23

Evaluate the polynomial P(x) = x^3 - 2x^2 + 5 at x = 2.

Accepted Answer

11

Answer

9

Answer

1

Answer

3

Question 24

Which property does NOT apply to polynomials?

Accepted Answer

Symmetric

Answer

Commutative under addition and multiplication

Answer

Associative under addition and multiplication

Answer

Distributive over addition and subtraction

Question 25

When performing an operation on two polynomials, one with a degree of 3 and the other with a degree of 2, which operation will result in a polynomial with a degree of 5?

Accepted Answer

Multiply

Answer

Add

Answer

Subtract

Answer

Divide

Question 26

Which property does NOT apply to polynomials?

Accepted Answer

A polynomial of degree n has exactly n distinct real roots

Answer

Closure under addition, subtraction, and multiplication

Answer

Commutativity and associativity of addition and multiplication

Answer

Distributivity of multiplication over addition and subtraction

Question 27

Which of the following expressions is a polynomial?

Accepted Answer

x^2 + 2x - 5

Answer

√x + 2x - 5

Answer

x^2 + 2x^-1 - 5

Question 28

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

Accepted Answer

4

Answer

2

Answer

-2

Answer

5

Question 29

Which of the following is NOT a characteristic of polynomials?

Accepted Answer

Contains logarithmic functions

Answer

Can be used to model real-world scenarios

Answer

Can be written as a sum of monomials

Answer

Has a finite number of terms

Question 30

Subtract (5x - 2) from (x^2 + 3x - 5)

Accepted Answer

x^2 - 8x + 3

Answer

x^2 + 8x - 3

Answer

5x + 3

Answer

-x^2 - 8x + 3

Question 31

Find the zeros of the polynomial: 2x^2 - 5x + 3

Accepted Answer

x = 1.5, x = 1

Answer

x = -1.5, x = -1

Answer

x = -2.5, x = -0.5

Answer

x = 2.5, x = 0.5

Question 32

Simplify the polynomial expression: (x² + 3x - 2) + (2x² - 5x + 4)

Accepted Answer

3x² - 2x + 2

Answer

x⁴ - 2x + 2

Answer

2x² - 2x + 6

Answer

x² + 8x - 6

Question 33

Divide the polynomial (x³ - 2x² + x - 2) by (x - 1). What is the quotient?

Accepted Answer

x² - x + 2

Answer

x² - 2x + 2

Answer

x³ - x² + x - 2

Answer

x² + x - 2

Question 34

What is the degree of the polynomial: 5x⁴ - 2x³ + x² - 3?

Accepted Answer

4

Answer

1

Answer

3

Answer

2

Question 35

What is the leading coefficient of the polynomial: -3x² + 5x - 1?

Accepted Answer

-3

Answer

5

Answer

-1

Answer

3

Question 36

What is the remainder when (x² - 3x + 2) is divided by (x - 2)?

Accepted Answer

0

Answer

-1

Answer

1

Question 37

Which of the following expressions is a polynomial?

Accepted Answer

3x² + 2x - 1

Answer

1/x² + 4

Answer

√x + 5

Answer

2x³ + 3x⁻¹ + 1

Question 38

What is the degree of the polynomial 5x⁴ - 2x³ + 7x - 1?

Accepted Answer

4

Answer

7

Answer

1

Answer

3

Question 39

Subtract the polynomial (x² - 4x + 3) from (3x² + 2x - 1).

Accepted Answer

2x² + 6x - 4

Answer

4x² - 2x + 2

Answer

2x² - 6x + 4

Answer

2x² - 2x - 4

Question 40

Multiply the polynomials: (x + 2)(x² - 3x + 1)

Accepted Answer

x³ - x² - 5x + 2

Answer

x³ - x² - 5x - 2

Answer

x³ - x² + 5x + 2

Answer

x³ + 2x² - 5x + 2

Question 41

Simplify the expression: (2x + 1)²

Accepted Answer

4x² + 4x + 1

Answer

4x² + 2x + 1

Answer

2x² + 4x + 1

Answer

4x² + 1

Question 42

What is the definition of a polynomial?

Accepted Answer

An expression composed of terms with constants and variables raised to non-negative integer powers, combined using addition, subtraction, and/or multiplication.

Answer

An equation with multiple terms and variables.

Answer

A function with a variable raised to a fractional exponent.

Answer

An expression with only one variable term.

Question 43

Simplify the polynomial expression: (2x² - 3x + 5) + (x² + 2x - 1)

Accepted Answer

3x² - x + 4

Answer

x² - x + 3

Answer

3x² + 5x + 4

Answer

2x² + x + 6

Question 44

Subtract the polynomial (3x² - 2x + 1) from (5x² + 4x - 3).

Accepted Answer

2x² + 6x - 4

Answer

8x² + 2x - 2

Answer

-2x² - 6x + 4

Answer

x² + 2x - 2

Question 45

Divide the polynomial (6x³ - 12x² + 10x) by the monomial (2x).

Accepted Answer

3x² - 6x + 5

Answer

x² + 2x + 5

Answer

x² - 2x + 5

Answer

3x² + 6x + 5

Question 46

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

Accepted Answer

3

Answer

4

Answer

2

Answer

5

Question 47

Evaluate the polynomial f(x) = x² - 2x + 3 when x = 2.

Accepted Answer

3

Answer

1

Answer

5

Answer

-1

Question 48

Which of the following is a factor of the polynomial: x³ - 2x² + 5x - 10?

Accepted Answer

(x - 2)

Answer

(x + 5)

Answer

(x + 2)

Answer

(x - 5)

Question 49

What is the standard form of the polynomial: 3x² - x + 2x³?

Accepted Answer

2x³ + 3x² - x

Answer

-x + 3x² + 2x³

Answer

x - 3x² + 2x³

Answer

3x² - x + 2x³

Question 50

Which of the following expressions is a polynomial?

Accepted Answer

3x² - 5x + 2

Answer

x³ + 1/x - 2

Answer

√x + 7x - 3

Answer

2x⁻¹ + 4x² - 1

Question 51

What is the degree of the polynomial 4x³ - 2x² + 5x - 1?

Accepted Answer

3

Answer

2

Answer

1

Answer

4

Question 52

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

Accepted Answer

3x² - x + 4

Answer

3x² + 7x + 4

Answer

x² - x + 4

Answer

3x² - 7x + 4

Question 53

A rectangular garden has a length of (2x + 3) meters and a width of (x - 1) meters. What is the area of the garden in terms of x?

Accepted Answer

2x² + x - 3

Answer

2x² - x - 3

Answer

2x² - x + 3