Master Polynomials: A Comprehensive Guide for Mechanical Engineering Students

Question 1

Find the degree of the polynomial (x² - 1)(x³ + 2x - 1).

Accepted Answer

6

Answer

5

Question 2

Find the remainder when the polynomial f(x) = 2x³ - 3x² + 5x - 1 is divided by x - 1. Use synthetic division.

Accepted Answer

4

Answer

6

Question 3

In a polynomial, which term is known as the constant term?

Accepted Answer

The term without any variables

Answer

The term with the highest exponent

Answer

The term with the lowest exponent

Answer

The sum of all the coefficients in the polynomial

Question 4

Multiply the polynomials ((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 5

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

Accepted Answer

(3, -5, 2, -1)

Answer

(-3, 5, -2, 1)

Answer

(3, 5, -2, -1)

Answer

(-3, -5, 2, 1)

Question 6

Write the polynomial (x^2 - 2xy + y^2) in factored form.

Accepted Answer

((x - y)^2)

Answer

((x + y)^2)

Answer

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

Answer

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

Question 7

Simplify: (x^2 + y^2)^2

Accepted Answer

x^4 + 2x^2y^2 + y^4

Answer

x^4 + y^4

Answer

x^4 - 2x^2y^2 + y^4

Answer

x^4 - y^4

Question 8

Identify the degree of the polynomial 3x⁴ - 2x³ + 5x² - 7.

Accepted Answer

4

Answer

2

Answer

3

Answer

5

Question 9

Find the roots of the polynomial x² - 5x + 6.

Accepted Answer

x = 2, x = 3

Answer

x = -2, x = -3

Answer

x = 2, x = -3

Answer

x = -2, x = 3

Question 10

Factor the polynomial 2x³ - 8x.

Accepted Answer

2x(x² - 4)

Answer

2(x³ - 4x)

Answer

x(2x² - 8)

Answer

2(x² - 4x)

Question 11

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

Accepted Answer

4

Answer

2

Answer

5

Question 12

In the polynomial 3x^4 - 2x^3 + 5x² - 7x - 6, identify the leading coefficient and constant term.

Accepted Answer

Leading coefficient: 3, Constant term: -6

Answer

Leading coefficient: -6, Constant term: 3

Answer

Leading coefficient: 3, Constant term: -2

Answer

Leading coefficient: -2, Constant term: 6

Question 13

Simplify the polynomial by combining like terms: (2x^2 + 3x - 5) + (x^2 - 2x + 4)

Accepted Answer

3x^2 + x - 1

Answer

x^2 + 5x - 1

Answer

3x^2 - x + 9

Question 14

Subtract the polynomial (3x^2 - 2x + 5) from (5x^2 + x - 3) using correct polynomial subtraction techniques:

Accepted Answer

2x^2 + 3x - 8

Answer

-2x^2 + 3x - 8

Answer

-2x^2 - 3x + 8

Answer

2x^2 - 3x + 2

Question 15

Find the degree of the polynomial 2x⁴ - 5x³ + 3x² - x + 1.

Accepted Answer

4

Answer

2

Answer

1

Answer

3

Question 16

Simplify the polynomial: (2x² + 3x - 5) + (x² - 2x + 7)

Accepted Answer

3x² + x + 2

Answer

4x² + 5x - 12

Answer

4x² - 5x + 2

Answer

3x² - x + 2

Question 17

Factor the polynomial: x² - 4

Accepted Answer

(x + 2)(x - 2)

Answer

(x² + 4)(x² - 4)

Answer

(x + 4)(x - 4)

Answer

(x - 2)²

Question 18

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

Accepted Answer

4

Answer

3

Answer

5

Answer

2

Question 19

What is the leading coefficient of the polynomial: -2x³ + 5x² - 3x + 1?

Accepted Answer

-2

Answer

-3

Answer

1

Answer

5

Question 20

What is the constant term of the polynomial: 2x³ - 4x² + 6x - 5?

Accepted Answer

-5

Answer

6

Answer

2

Answer

-4

Question 21

Is the polynomial x⁴ - 2x² + 1 a perfect square trinomial?

Accepted Answer

Yes

Answer

No

Answer

None of the above

Answer

Cannot be determined

Question 22

Which of the following expressions is a polynomial?

Accepted Answer

x³ - 2x² + 5x - 1

Answer

sin(x)

Answer

x⁻²

Answer

log(x)

Question 23

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

Accepted Answer

4

Answer

5

Answer

3

Answer

2

Question 24

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

Accepted Answer

x² + x - 6

Answer

x² + 5x - 6

Answer

x² - x - 6

Answer

2x² + x - 6

Question 25

Factor the polynomial completely: x² - 4

Accepted Answer

(x - 2)(x + 2)

Answer

Cannot be factored

Answer

(x - 4)(x + 4)

Answer

(x - 1)(x + 3)

Question 26

How many real roots does the polynomial x⁴ - 2x² + 1 have?

Accepted Answer

2

Answer

0

Answer

3

Answer

1

Question 27

What is the x-coordinate of a local minimum of the polynomial function f(x) = x³ - 3x² + 2x + 1?

Accepted Answer

1

Answer

-1

Answer

0

Answer

2

Question 28

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

Accepted Answer

4

Answer

3

Answer

1

Answer

2

Question 29

Identify the polynomial with a leading coefficient of 2 and a constant term of -3:

Accepted Answer

2x³ - 5x² + 4x - 3

Answer

3x² - 2x + 5

Answer

-3x⁴ + 2x² - 1

Answer

x³ + 2x - 3

Question 30

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

Accepted Answer

3x² + x + 4

Answer

2x² + x + 4

Answer

3x² + 5x + 4

Question 31

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

Accepted Answer

x² - x - 6

Answer

x² + x - 6

Answer

x² + x + 6

Answer

x² - x + 6

Question 32

Divide (x³ - 2x² + 5x - 3) by (x - 1) using long division or synthetic division:

Accepted Answer

x² - x + 4 + 1/(x - 1)

Answer

x² + x + 4 + 1/(x - 1)

Answer

x² - x + 4 - 1/(x - 1)

Answer

x² + x + 4 - 1/(x - 1)

Question 33

Evaluate the polynomial 2x³ - 5x + 1 when x = -2:

Accepted Answer

-19

Answer

19

Answer

15

Answer

-15

Question 34

Determine the roots of the quadratic equation x² - 5x + 6 = 0:

Accepted Answer

x = 2, 3

Answer

x = -2, 3

Answer

x = 2, -3

Answer

x = -2, -3

Question 35

Identify a factor of the polynomial x³ + 2x² - 5x - 6:

Accepted Answer

x + 1

Answer

x - 1

Answer

x - 2

Answer

x + 2

Question 36

Apply the Remainder Theorem to find the remainder when the polynomial 3x⁴ - 2x² + 5x - 1 is divided by (x + 2):

Accepted Answer

27

Answer

17

Answer

-27

Answer

-17

Question 37

Calculate the area of a rectangular garden with a length of (x + 5) meters and a width of (x - 2) meters:

Accepted Answer

x² + 3x - 10

Answer

x² - 3x - 10

Answer

x² - 3x + 10

Answer

x² + 3x + 10

Question 38

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

Accepted Answer

4

Answer

5

Answer

2

Answer

3

Question 39

Which of the following is a monomial?

Accepted Answer

7x^3

Answer

x^2 + 2x

Answer

2x^4 - x^3 + 4x

Answer

5x^2 - 3x + 1

Question 40

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

Accepted Answer

6x + 2

Answer

6x - 2

Answer

2x + 2

Answer

8x + 2

Question 41

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

Accepted Answer

x^2 - x - 6

Answer

x^2 - x + 6

Answer

x^2 + x + 6

Answer

x^2 + x - 6

Question 42

Divide (x^3 + 2x^2 - 5x - 6) by (x + 2).

Accepted Answer

x^2 - 5

Answer

x^2 + 5

Answer

x^2 + 2x + 1

Answer

x^2 - 2x - 1

Question 43

Find the value of the polynomial 2x^3 - 4x^2 + 3x - 1 when x = 2.

Accepted Answer

11

Answer

7

Answer

1

Answer

5

Question 44

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 45

What is the solution to the equation x^2 - 5x + 6 = 0?

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

Question 46

The polynomial 2x^2 + 5x - 3 can be factored as:

Accepted Answer

(2x - 1)(x + 3)

Answer

(2x + 1)(x - 3)

Answer

(x - 1)(2x + 3)

Answer

(x + 1)(2x - 3)

Question 47

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

Accepted Answer

x^2 + 3x - 10 square meters

Answer

x^2 + 3x + 10 square meters

Answer

x^2 - 3x - 10 square meters

Answer

x^2 - 3x + 10 square meters