Polynomials: Master the Fundamentals of Polynomial Operations

Question 1

Which statement is NOT a property of polynomials?

Accepted Answer

The coefficients of polynomials must be integers.

Answer

Polynomials are closed under addition and subtraction.

Answer

Polynomials are closed under multiplication.

Answer

Polynomials can have any number of terms.

Question 2

Find the sum of the polynomials (2x^3 - 5x^2 + 3x - 1) and (x^3 + 2x^2 - 4x + 5).

Accepted Answer

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

Answer

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

Answer

3x^3 - 7x^2 - x + 4

Answer

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

Question 3

Divide the polynomial 2x^3 - 5x^2 + 3x - 1 by x - 2.

Accepted Answer

2x^2 + x - 1

Answer

2x^2 + 2x + 1

Answer

2x^2 - x + 1

Answer

2x^2 - 2x + 1

Question 4

Factor the expression x^3 - 2x^2 - 5x + 6 into (a + b) form.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 5

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

Accepted Answer

x³ + 6x² + 12x + 8

Answer

x³ + 8

Answer

x³ + 6x² + 12x

Answer

x³ + 12x² + 8

Question 6

Factor the following polynomial by grouping: x³ - 4x² + x - 4

Accepted Answer

(x - 4)(x² + x + 1)

Answer

(x + 4)(x² - x + 1)

Answer

(x - 2)(x² + 2x + 2)

Answer

(x + 2)(x² - 2x + 2)

Question 7

Determine the roots of the polynomial f(x) = x³ - 3x² + 2x - 1

Accepted Answer

x = -1, x = 0, x = 1

Answer

x = 0, x = 1, x = 2

Answer

x = -2, x = 0, x = 2

Answer

x = 1, x = 2, x = 3

Question 8

Calculate the product of (x - 2) and (x² + 2x + 4):

Accepted Answer

x³ + 2x² + 4x - 8

Answer

x³ - 8

Answer

x³ - 4x² + 8

Answer

x³ + 2x² - 4x - 8

Question 9

Simplify the expression (x + y)² - (x - y)²:

Accepted Answer

4xy

Answer

2x

Answer

2y

Answer

8xy

Question 10

Which characteristic is unique to a quadratic polynomial?

Accepted Answer

Has a parabolic graph

Answer

Has a degree of 2

Answer

Symmetrical about the x-axis

Answer

Has two x-intercepts

Question 11

Which scenario best represents an application of polynomials?

Accepted Answer

Modelling the motion of a projectile

Answer

Calculating the volume of a sphere

Answer

Determining the slope of a line

Answer

Simplifying trigonometric identities

Question 12

In the polynomial 6x^4 - 3x^2 + 7x - 5, what is the coefficient of the x^2 term?

Accepted Answer

-3

Answer

3

Answer

6

Answer

7

Question 13

Factorize the polynomial: x^2 - 16

Accepted Answer

(x - 4)(x + 4)

Answer

(x - 4)^2

Answer

(x + 4)^2

Answer

Cannot be factorized

Question 14

Simplify the expression: (3x^2y^3 - 6xy^2 + 9xy) / (3xy)

Accepted Answer

xy^2 - 2y + 3

Answer

x^2y^2 - 2y + 3

Answer

3x - 2 + 3y

Answer

3x^2 - 2y + 3y^2

Question 15

Identify the polynomial from the following expressions:

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 1/x^2

Answer

sin(x)

Answer

2^(3x)

Question 16

Factorize the following polynomial: x^2 - 9

Accepted Answer

(x - 3)(x + 3)

Answer

(x - 9)

Answer

Cannot be factorised

Answer

(x - 3)^2

Question 17

Find the quotient when the polynomial 2x^3 - 4x^2 + 6x - 8 is divided by x - 2.

Accepted Answer

2x^2 + 2x + 4

Answer

2x^2

Answer

2x^2 - 2x + 4

Answer

x^2 - 2x + 4

Question 18

Factor the expression: x^2 - 9

Accepted Answer

(x - 3)(x + 3)

Answer

(x - 9)(x + 9)

Answer

(x - 1)(x + 1)

Answer

This expression cannot be factored.

Question 19

Simplify the expression: (4x^2y - 2xy^2) / 2xy

Accepted Answer

2x - y

Answer

x - y

Answer

2xy - y^2

Answer

x - 2y^2

Question 20

Simplify the polynomial expression:** (2x - 1)^2

Accepted Answer

4x^2 - 4x + 1

Answer

4x^2 - 1

Answer

2x^2 - 4x + 1

Answer

4x^2 + 4x + 1

Question 21

Factor the polynomial:** x^2 - 9

Accepted Answer

(x + 3)(x - 3)

Answer

(x - 3)^2

Answer

(x + 9)(x - 9)

Answer

(x - 9)^2

Question 22

Factor the polynomial:** 4x^2 - 25

Accepted Answer

(2x + 5)(2x - 5)

Answer

(4x + 5)(x - 5)

Answer

(2x - 5)^2

Answer

(4x - 5)^2

Question 23

Simplify the expression (x² + 2x - 3) / (x² - 9).

Accepted Answer

1 / (x - 3)

Answer

1 / (x + 3)

Answer

-1 / (x + 3)

Answer

-1 / (x - 3)

Question 24

Factorise the polynomial: x² - 4.

Accepted Answer

(x - 2)(x + 2)

Answer

(x + 2)²

Answer

(x - 2)²

Answer

(x + 4)(x - 4)

Question 25

Find the quotient and remainder when x³ - 2x² + x - 1 is divided by x - 1.

Accepted Answer

Quotient: x² - x + 1; Remainder: 0

Answer

Quotient: x² + x + 1; Remainder: 0

Answer

Quotient: x² - x - 1; Remainder: 0

Answer

Quotient: x² + x - 1; Remainder: 1

Question 26

Which of the following is a factor of x⁴ - 16?

Accepted Answer

x² - 4

Answer

x - 2

Answer

x + 2

Answer

x² + 4

Question 27

Simplify the expression: (x + 3)² - (x - 2)(x + 5).

Accepted Answer

x² + 17x - 11

Answer

x² - 17x - 11

Answer

x² + 17x + 11

Answer

x² - 17x + 11

Question 28

Which of the following is a polynomial of degree 4?

Accepted Answer

x⁴ - x³ + 2x² - 5x + 1

Answer

x² - 2x + 1

Answer

x³ - 2x² + x + 3

Answer

x³ - 3x² + 2x + 4

Question 29

Factorise the polynomial: x³ - 2x² + x - 2.

Accepted Answer

(x - 1)(x² + x - 2)

Answer

(x + 1)(x² - x - 2)

Answer

(x - 2)(x² + x - 1)

Answer

(x + 2)(x² - x - 1)

Question 30

Find the remainder when the polynomial x³ - 2x² + x + 1 is divided by (x - 1).

Accepted Answer

3

Answer

-3

Answer

1

Answer

-1

Question 31

Which of the following is a factor of the polynomial: x^3 - 8?

Accepted Answer

x - 2

Answer

x + 2

Answer

x - 1

Answer

x + 1

Question 32

Find the remainder when the polynomial: **x³ - 2x² + x - 2** is divided by **(x - 1)**

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 33

What is the quotient of **(x³ - 8)** divided by **(x - 2)**?

Accepted Answer

x² + 2x + 4

Answer

x² - 2x - 4

Answer

x² + 2x - 4

Answer

x² - 2x + 4

Question 34

Determine the degree of the polynomial (2x^4 - 3x^2 + 5x - 1).

Accepted Answer

4

Answer

2

Answer

3

Answer

5

Question 35

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

Accepted Answer

x^2 - 5x + 6

Answer

x^2 + x - 6

Answer

x^2 - x - 6

Answer

x^2 + 5x + 6

Question 36

Factor the polynomial: x^2 - 4.

Accepted Answer

(x + 2)(x - 2)

Answer

(x - 2)^2

Answer

(x + 4)(x - 4)

Answer

The polynomial cannot be factored.

Question 37

Which of the following is the sum of (x - 2)(x + 3) and (x + 1)(x - 4)?

Accepted Answer

x² - 5x - 6

Answer

x² + 5x - 6

Answer

x² - x - 6

Answer

x² + x - 6

Question 38

What is the product of (x - 3) and (x² + 3x + 9)?

Accepted Answer

x³ - 9x² - 27x + 27

Answer

x³

Answer

x³ - 27

Answer

x³ - 9x² + 27x - 27

Question 39

Factor the polynomial x^2 - 9 by identifying the difference of squares:

Accepted Answer

(x + 3)(x - 3)

Answer

The polynomial cannot be factored over real numbers

Answer

(x - 3)^2

Answer

(x + 9)(x - 9)

Question 40

Determine whether the polynomial x^2 + 2x + 1 is a quadratic polynomial:

Accepted Answer

Yes

Answer

It is not a polynomial

Answer

It is a quartic polynomial

Answer

It is a cubic polynomial

Question 41

Factor the polynomial: x^2 - 9

Accepted Answer

(x + 3)(x - 3)

Answer

(x + 3)^2

Answer

(x - 3)^2

Answer

(x + 9)(x - 9)

Question 42

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

Accepted Answer

5

Answer

7

Answer

2

Answer

3

Question 43

Factor the polynomial: x^4 - 16

Accepted Answer

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

Answer

(x + 4)(x - 4)

Answer

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

Answer

(x - 4)^2

Question 44

Find the zeros of the polynomial: x^2 - 5x + 6

Accepted Answer

2, 3

Answer

-1, 6

Answer

1, -6

Answer

-2, -3

Question 45

Identify the polynomial among the given expressions:

Accepted Answer

2x^3 - 5x + 1

Answer

e^x - 5x + 1 (Exponential function)

Answer

2x^(1/2) - 5x + 1 (Radical expression)

Answer

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

Question 46

Factor the polynomial x^2 - 9 using the difference of squares formula:

Accepted Answer

(x - 3)(x + 3)

Answer

(x - 3)^2 (Incorrect, it should be x + 3)

Answer

(x - 9)(x + 9) (Incorrect difference of squares)

Answer

Cannot be factored (Incorrect, it can be factored)

Question 47

Factor the polynomial x^2 + 6x + 8 using the perfect square trinomial formula:

Accepted Answer

(x + 2)(x + 4)

Answer

Cannot be factored (Incorrect, it can be factored)

Answer

(x + 6)(x + 8) (Incorrect, coefficients of x should be equal)

Answer

(x - 2)(x - 4) (Incorrect perfect square trinomial)

Question 48

Factor the polynomial x^3 - 27 using the sum or difference of cubes formula:

Accepted Answer

(x - 3)(x^2 + 3x + 9)

Answer

(x - 9)(x^2 + 9) (Incorrect, it should be x^2 + 3x + 9)

Answer

(x + 3)(x^2 - 3x + 9) (Incorrect sum or difference of cubes)

Question 49

Perform long division to simplify the expression: (2x^3 - 5x^2 + 3x) ÷ (x - 2)

Accepted Answer

2x^2 + x - 1

Answer

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

Answer

2x^3 + x - 1 (Incorrect result)

Answer

x^2 - 5x + 3 (Incorrect result)

Question 50

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

Accepted Answer

4

Answer

1 (Incorrect, constant term is not the degree)

Answer

3 (Incorrect, highest exponent is 4)

Answer

2 (Incorrect, highest exponent is 4)

Question 51

Using the Remainder Theorem, find the value of k such that when the polynomial x^3 + 2x^2 - 5x + k is divided by (x-2), the remainder is 7:

Accepted Answer

-3

Answer

11 (Incorrect value of k)

Answer

-11 (Incorrect value of k)

Answer

3 (Incorrect value of k)

Question 52

Identify a factor of the quadratic polynomial x^2 - 7x + 12:

Accepted Answer

(x-3)

Answer

(x+4) (Incorrect factor)

Answer

(x+3) (Incorrect factor)

Answer

(x-2) (Incorrect factor)

Question 53

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

Accepted Answer

3x^2 + 2x - 1

Answer

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

Answer

3x^2 - 8x + 3 (Incorrect simplification)

Answer

3x^2 + 8x - 3 (Incorrect simplification)

Question 54

Multiply the polynomials: (x-3)(x^2 + 3x + 9)

Accepted Answer

x^3 - 27

Answer

x^3 - 9x - 27 (Incorrect product)

Answer

x^3 + 27 (Incorrect product)

Question 55

Factor completely: $4x^2 - 9y^2$

Accepted Answer

(2x + 3y)(2x - 3y)

Answer

(2x - 3y)^2

Answer

(2x + 3y)^2

Answer

(4x + 9y)(4x - 9y)

Question 56

Which of the following is a polynomial equation with roots -2, 1, and 3?

Accepted Answer

(x+2)(x-1)(x-3) = 0

Answer

(x+2)(x-1)(x+3) = 0

Answer

(x-2)(x+1)(x+3) = 0

Answer

(x-2)(x+1)(x-3) = 0

Question 57

The polynomial $f(x)$ has a factor of (x-2) and a double root at x = -1. Which of the following could be the equation for f(x)?

Accepted Answer

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

Answer

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

Answer

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

Question 58

What is the quotient when $2x^3 - 7x^2 + 5x - 1$ is divided by (x-2)?

Accepted Answer

2x^2 - 3x - 1

Answer

2x^2 + 3x - 1

Answer

2x^2 + 3x + 1

Answer

2x^2 - 3x + 1

Question 59

A rectangular box has a volume represented by the polynomial $x^3 - 6x^2 + 11x - 6$. If the height of the box is (x-1), what are the possible dimensions for the length and width?

Accepted Answer

(x-2) and (x-3)

Answer

(x+2) and (x+3)

Answer

(x-2) and (x+3)

Answer

(x+2) and (x-3)