Polynomials: Operations, Factoring, and Remainder Theorem

Question 1

Which of the following is NOT a factor of the polynomial x⁴ - 1?

Accepted Answer

x² + 1

Answer

x + 1

Answer

x - 1

Answer

x² - 1

Question 2

Which operation is not typically performed on polynomials?

Accepted Answer

Integration

Answer

Addition

Answer

Subtraction

Answer

Multiplication

Question 3

Find the product of (x + 2) and (x - 3).

Accepted Answer

x^2 - x - 6

Answer

x^2 + x - 6

Answer

x^2 - 5x + 6

Answer

x^2 + 5x + 6

Question 4

Which value of x is NOT a zero of the polynomial x³ - x² - 2x + 2?

Accepted Answer

x = 2

Answer

x = 1

Answer

x = -1

Answer

x = -2

Question 5

Determine the degree of the polynomial 3x⁵ - 2x³ + 4x.

Accepted Answer

5

Answer

3

Answer

4

Answer

6

Question 6

Simplify the expression (4x²y)³.

Accepted Answer

64x⁶y³

Answer

64x⁵y²

Answer

64x⁶y²

Answer

64x⁵y³

Question 7

Identify the standard form of the polynomial 5x² - 3x + 1.

Accepted Answer

5x² - 3x + 1

Answer

x² - 3x + 5

Answer

3x - 5x² + 1

Answer

1 + 3x - 5x²

Question 8

Consider the polynomial equation y = x² - 4x + 3. Identify the shape of the graph represented by this equation and provide a clear explanation for your choice.

Accepted Answer

Parabola

Answer

Circle

Answer

Hyperbola

Answer

Ellipse

Question 9

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

Accepted Answer

(x - 2)^2

Answer

(x + 2)^2

Answer

(x - 4)(x + 4)

Answer

(x + 4)(x + 4)

Question 10

Find the zeros of the polynomial x^3 + 2x^2 - 5x - 6.

Accepted Answer

-1, 2, 3

Answer

-1, -2, -3

Answer

1, 2, 3

Answer

1, -2, -3

Question 11

Which of the following is NOT a property of polynomials?

Accepted Answer

They have a finite number of terms.

Answer

They are closed under addition and subtraction.

Answer

They are closed under multiplication.

Answer

They are continuous functions.

Question 12

Which of the following polynomials is a perfect square?

Accepted Answer

x^2 - 4x + 4

Answer

x^2 + 4x + 4

Answer

x^2 - 2x + 1

Answer

x^2 + 2x + 1

Question 13

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

Accepted Answer

4

Answer

2

Answer

3

Answer

5

Question 14

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

Accepted Answer

x^2 + x - 6

Answer

x^2 + 5x - 4

Answer

3x^2 + 5x - 6

Answer

x^2 - x + 4

Question 15

Given f(x) = 2x^3 + x^2 - 3x + 2 and g(x) = x^3 - 2x^2 + 5x - 1, what is f(x) - g(x)?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 16

Factor the polynomial f(x) = x^2 - 4x + 4.

Accepted Answer

(x - 2)^2

Answer

(x + 2)^2

Answer

(x - 4)(x + 1)

Answer

(x + 4)(x - 1)

Question 17

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

Accepted Answer

x^2 - 2x + 1

Answer

x^2 + 2x + 1

Answer

x^2 - 2x - 1

Answer

x^2 + 2x - 1

Question 18

Which of the following is a polynomial?

Accepted Answer

x^2 + 2x - 3

Answer

e^x + ln(x)

Answer

sin(x) + cos(x)

Answer

2x^(1/2) + 3

Question 19

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

Accepted Answer

3

Answer

1

Answer

2

Answer

5

Question 20

Classify the polynomial S(x) = 2x^4 - 3x^2 + 5 based on its degree.

Accepted Answer

Quartic polynomial

Answer

Quadratic polynomial

Answer

Linear polynomial

Answer

Cubic polynomial

Question 21

Which of the following is a root of the polynomial T(x) = x^2 - 5x + 6?

Accepted Answer

3

Answer

-3

Answer

-2

Answer

2

Question 22

Which of the following is a polynomial expression?

Accepted Answer

x^2 + 2x - 5

Answer

sin(x)

Answer

e^x

Answer

1/(x+2)

Question 23

Find the degree of the polynomial P(x) = 3x^4 + 2x^2 - 5x + 1

Accepted Answer

4

Answer

2

Answer

3

Answer

5

Question 24

Factor the following polynomial: x^2 - 4x + 4

Accepted Answer

(x-2)^2

Answer

(x-2)(x+2)

Answer

(x+2)(x-4)

Answer

(x+2)^2

Question 25

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

Accepted Answer

4

Answer

3

Answer

5

Answer

2

Question 26

Expand the polynomial (x + 3)^2

Accepted Answer

x^2 + 6x + 9

Answer

x^2+3x+9

Answer

x^2-6x+9

Answer

x^2-3x+9

Question 27

Identify the polynomial from the given options:

Accepted Answer

x^2 + 2x - 3

Answer

sin(x)

Answer

1 / (x - 2)

Answer

log(x)

Question 28

Factor the polynomial x^2 - 9.

Accepted Answer

(x + 3)(x - 3)

Answer

(x + 9)(x - 9)

Answer

(x + 3)^2

Question 29

Apply the remainder theorem to find the remainder when x^3 - 2x^2 + 5x - 7 is divided by (x - 1).

Accepted Answer

1

Answer

0

Answer

-1

Question 30

Determine the leading coefficient of the polynomial: -3x⁴ + 5x³ - 2x² + 1

Accepted Answer

-3

Answer

-2

Answer

1

Answer

5

Question 31

Classify the polynomial: x² + 2xy + y²

Accepted Answer

Trinomial

Answer

Quadratic

Answer

Binomial

Answer

Cubic

Question 32

Determine the degree of the polynomial P(x) = x³ - 2x² + 3x - 5:

Accepted Answer

3

Answer

1

Answer

4

Answer

2

Question 33

Identify the leading coefficient of the polynomial P(x) = -2x³ + 5x² - 3x + 1:

Accepted Answer

-2

Answer

3

Answer

5

Answer

1

Question 34

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

Accepted Answer

4

Answer

3

Answer

1

Answer

2

Question 35

What is the sum of the polynomials (2x² + 3x - 1) and (x² - 2x + 5)?

Accepted Answer

3x² + x + 4

Answer

x² + x + 4

Answer

3x² + x - 6

Answer

3x² + 5x + 4

Question 36

Which of the following is a quadratic polynomial?

Accepted Answer

2x² + 3x - 1

Answer

4

Answer

3x³ - 2x² + 5x

Answer

5x + 2

Question 37

What is the value of the polynomial 2x³ - 5x + 1 when x = 2?

Accepted Answer

9

Answer

1

Answer

11

Answer

-7

Question 38

Which of the following polynomials has a leading coefficient of 3?

Accepted Answer

3x⁴ - 2x² + 5x - 1

Answer

x² - 4x + 3

Answer

2x³ - 5x² + 1

Answer

5x³ + 2x - 1

Question 39

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

Accepted Answer

10x + 5

Answer

2x + 5

Answer

2x - 5

Answer

10x - 5

Question 40

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

Accepted Answer

4

Answer

3

Answer

5

Answer

2

Question 41

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

Accepted Answer

7x + 2

Answer

10x - 3

Answer

10x + 2

Answer

7x - 4

Question 42

What is the product of (x + 2) and (x - 3)?

Accepted Answer

x^2 - x - 6

Answer

x^2 - 5x - 6

Answer

x^2 + 5x - 6

Answer

x^2 + x - 6

Question 43

Divide the polynomial 2x^3 + 5x^2 - 4x + 1 by x + 2.

Accepted Answer

2x^2 + x - 6 + 13/(x+2)

Answer

2x^2 - x + 6 + 13/(x+2)

Answer

2x^2 + x + 6 + 13/(x+2)

Answer

2x^2 - x - 6 + 13/(x+2)

Question 44

Factor the polynomial x^2 - 9 completely.

Accepted Answer

(x + 3)(x - 3)

Answer

(x - 9)(x + 1)

Answer

(x + 3)(x + 3)

Answer

(x + 9)(x - 1)

Question 45

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

Accepted Answer

3

Answer

0

Answer

-1

Answer

1

Question 46

Which of the following is a factor of the polynomial x^3 + 2x^2 - 5x - 6?

Accepted Answer

x + 3

Answer

x + 1

Answer

x - 3

Answer

x - 1

Question 47

Find the value of k for which the polynomial x^2 + kx - 6 is divisible by x - 2.

Accepted Answer

7

Answer

-1

Answer

3

Answer

5

Question 48

What is the simplified form of (x^2 - 4)/(x + 2) when x ≠ -2?

Accepted Answer

x - 2

Answer

x^2 - 2

Answer

x + 2

Answer

x + 4

Question 49

If the polynomial p(x) = x^3 + ax^2 + bx - 6 has factors (x + 1) and (x - 2), find the values of a and b.

Accepted Answer

a = 1, b = -10

Answer

a = -1, b = -10

Answer

a = -1, b = 10

Answer

a = 1, b = 10

Question 50

Which of the following is a polynomial?

Accepted Answer

x^2 + 2x + 1

Answer

sqrt(x)

Answer

1/x

Answer

sin(x)

Question 51

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

Accepted Answer

3x^2 - x - 4

Answer

x^2 - 5x - 4

Answer

2x^2 + 5x - 4

Answer

x^2 + x - 4

Question 52

Multiply the following polynomials: (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 53

Factor the following polynomial: x^2 - 4

Accepted Answer

(x - 2)(x + 2)

Answer

(x - 1)(x + 4)

Answer

(x - 4)(x + 1)

Answer

(x - 2)(x - 2)

Question 54

Use the remainder theorem to find the remainder when x^3 - 2x^2 + x - 1 is divided by x - 1.

Accepted Answer

-1

Answer

1

Answer

2

Answer

0

Question 55

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

Accepted Answer

2

Answer

-1

Answer

0

Answer

1

Question 56

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

Accepted Answer

4

Answer

3

Answer

2

Question 57

Which of the following is a leading coefficient of the polynomial x^3 - 2x^2 + 3x - 1?

Accepted Answer

1

Answer

3

Answer

-1

Answer

-2

Question 58

Determine if the polynomial x^2 - 4x + 4 is a perfect square trinomial.

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Answer

None of the above

Question 59

Simplify the following expression using synthetic division: (x^3 - 2x^2 + x - 1) ÷ (x - 1)

Accepted Answer

x^2 - x + 1

Answer

x^2 + x - 1

Answer

x^2 - x - 1

Answer

x^2 - 2x + 1