Master Factoring Polynomials: Enhance Your Algebra Skills

Question 1

Find the quadratic formula and describe its purpose:

Accepted Answer

x = (-b ± √(b² - 4ac)) / 2a, where 'a', 'b', 'c' are coefficients of a quadratic equation; Used to solve quadratic equations

Answer

x = (-b ± √(b² + 4ac)) / 2a

Answer

x = (b ± √(b² - 4ac)) / 2a

Answer

x = (b ± √(b² + 4ac)) / 2a, where 'a', 'b', 'c' are coefficients of a linear equation

Question 2

Factor the polynomial x³ - 8:

Accepted Answer

(x - 2)(x² + 2x + 4)

Answer

(x + 2)(x² - 2x + 4)

Answer

(x - 4)(x² + 4x + 16)

Answer

(x + 4)(x² - 4x + 16)

Question 3

Explain the concept of synthetic division:

Accepted Answer

A method for dividing a polynomial by a linear factor (x - a)

Answer

A method for dividing a polynomial by a quadratic factor (x² - a²)

Answer

A method for dividing a polynomial by a cubic factor (x³ - a³)

Answer

A method for dividing a polynomial by any type of factor

Question 4

Which of the following is NOT a method used for factoring polynomials?

Accepted Answer

Using the remainder theorem

Answer

Factoring by grouping

Answer

Trial and error

Answer

Quadratic formula

Question 5

Which of the following is the factored form of the polynomial x³ - 8?

Accepted Answer

(x - 2)(x² + 2x + 4)

Answer

(x + 2)(x² - 2x + 4)

Answer

(x - 2)(x² - 2x - 4)

Answer

(x + 2)(x² + 2x - 4)

Question 6

Factor the polynomial x² + 4x + 3 into irreducible factors.

Accepted Answer

The polynomial is irreducible.

Answer

(x + 1)(x + 3)

Answer

(x - 1)(x + 3)

Answer

(x + 3)(x + 1)

Question 7

Which of the following is NOT a method for factoring polynomials?

Accepted Answer

Substitution

Answer

Factoring by grouping

Answer

Trial and error

Answer

Quadratic formula

Question 8

Factor the polynomial: x² - 5x + 6 using the quadratic formula.

Accepted Answer

(x - 2)(x - 3)

Answer

(x + 2)(x - 3)

Answer

(x - 3)(x - 2)

Answer

(x + 3)(x - 2)

Question 9

Which of the following is a factor of the polynomial x³ - 27?

Accepted Answer

x - 3

Answer

x + 3

Answer

x² + 9

Answer

x² - 3

Question 10

Find the quadratic formula for solving the general quadratic equation ax² + bx + c = 0.

Accepted Answer

x = (-b ± √(b² - 4ac)) / 2a

Answer

x = (-b ± √(b² + 4ac)) / 2a

Answer

x = (-b - √(b² - 4ac)) / 2a

Answer

x = (-b + √(b² - 4ac)) / 2a

Question 11

Which of the following is NOT a technique for factoring polynomials?

Accepted Answer

Synthetic division

Answer

Grouping

Answer

Trinomial factoring

Answer

Quadratic formula

Question 12

Which of the following methods is not used for factoring polynomials?

Accepted Answer

Long division

Answer

Factoring by grouping

Answer

Trial and error

Answer

Quadratic formula

Question 13

Factor the polynomial x^2 - 4x + 4 using factoring by grouping.

Accepted Answer

(x - 2)^2

Answer

(x + 2)^2

Answer

(x + 4)^2

Answer

Cannot be factored by grouping

Question 14

Factor the polynomial 2x^4 - 16x^2 + 32 by grouping.

Accepted Answer

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

Answer

2(x^2 - 8)^2

Answer

2(x^2 + 8)^2

Answer

Cannot be factored by grouping

Question 15

Given a quadratic polynomial with roots at x = 2 and x = -5, write the factors of the polynomial.

Accepted Answer

(x - 2)(x + 5)

Answer

(x + 2)(x - 5)

Answer

(x - 2)(x - 5)

Answer

(x + 2)(x + 5)

Question 16

Factor the polynomial: x^2 - 5x + 6

Accepted Answer

(x - 2)(x - 3)

Answer

(x + 2)(x - 3)

Answer

(x - 1)(x - 6)

Answer

(x + 1)(x - 6)

Question 17

Identify the technique used to factor the polynomial **x⁴ - y⁴**.

Accepted Answer

Difference of squares

Answer

Quadratic formula

Answer

Factoring by grouping

Answer

Trial and error

Question 18

Factor the polynomial **x² - 9** using the method of difference of squares.

Accepted Answer

(x + 3)(x - 3)

Answer

(x + 9)(x - 1)

Answer

(x - 3)²

Question 19

Factor the polynomial **x² - 5x + 6** using the quadratic formula, showing the steps involved.

Accepted Answer

(x - 2)(x - 3)

Answer

(x + 2)(x + 3)

Answer

(x - 1)(x - 6)

Question 20

Factor the polynomial **x³ - 8** using the method of difference of cubes.

Accepted Answer

(x - 2)(x² + 2x + 4)

Answer

Cannot be factored

Answer

(x + 2)(x² - 4)

Answer

(x - 2)³

Question 21

Factor the polynomial **x² - 6x + 8** using factoring by grouping.

Accepted Answer

(x - 2)(x - 4)

Answer

(x + 2)(x + 4)

Answer

(x - 1)(x - 7)

Question 22

Factor the polynomial x² + 5x + 6 into the product of two binomials.

Accepted Answer

(x + 2)(x + 3)

Answer

(x + 1)(x + 6)

Answer

(x - 2)(x - 3)

Answer

(x + 1)(x - 6)

Question 23

Factor the polynomial 4x² - 9 into two binomials.

Accepted Answer

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

Answer

(2x + 9)(2x - 1)

Answer

(4x + 3)(x - 3)

Answer

(4x + 1)(x - 9)

Question 24

Fully factor the expression 3x³ + 6x² - 9x.

Accepted Answer

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

Answer

x(3x² + 6x - 9)

Answer

3x(x² + 2x - 3)

Answer

3(x³ + 2x² - 3x)

Question 25

Factor the polynomial 2x² + 7x + 3 into the product of two binomials.

Accepted Answer

(2x + 1)(x + 3)

Answer

(2x - 1)(x - 3)

Answer

(2x - 3)(x - 1)

Answer

(2x + 3)(x + 1)

Question 26

Find the solutions to the equation x² - 5x + 6 = 0 by factoring.

Accepted Answer

x = 2, x = 3

Answer

x = 1, x = 6

Answer

x = -1, x = -6

Answer

x = -2, x = -3

Question 27

Fully factor the polynomial 2x³ - 16x.

Accepted Answer

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

Answer

x(2x² - 16)

Answer

2x(x² - 8)

Question 28

Factor the polynomial 9x² - 12x + 4 into the product of two binomials.

Accepted Answer

(3x - 2)²

Answer

(3x - 4)(3x + 1)

Answer

(3x + 2)²

Answer

(9x - 2)(x - 2)

Question 29

Find the roots of the equation 3x² - 10x + 8 = 0 using the quadratic formula.

Accepted Answer

x = 2, x = 4/3

Answer

x = -2, x = -4/3

Answer

x = 1, x = 8/3

Answer

x = -1, x = -8/3

Question 30

Use the trial and error method to factor the polynomial 2x² - 5x + 2 into the product of two binomials.

Accepted Answer

(2x - 1)(x - 2)

Answer

(2x - 2)(x - 1)

Answer

(2x + 1)(x - 2)

Answer

2x² - 5x + 2 is prime

Question 31

Factor the polynomial x² + 5x + 6 into the product of two binomials.

Accepted Answer

(x + 2)(x + 3)

Answer

(x + 3)(x - 2)

Answer

The polynomial is already factored

Answer

(x - 2)(x - 3)

Question 32

Factor the quadratic expression  x² - 6x + 8 using the quadratic formula.

Accepted Answer

(x - 2)(x - 4)

Answer

(x - 4)(x + 2)

Answer

The expression cannot be factored.

Answer

(x + 2)(x + 4)

Question 33

Factor the trinomial 6x² - 13x + 6 completely.

Accepted Answer

(3x - 2)(2x - 3)

Answer

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

Answer

(3x + 2)(2x + 3)

Answer

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

Question 34

Identify a factor of the polynomial x³ - 8.

Accepted Answer

x - 2

Answer

x²

Answer

x⁴

Answer

x + 2

Question 35

Factor the polynomial x⁴ - y⁴ completely.

Accepted Answer

(x + y)(x - y)(x² + y²)

Answer

(x + y)⁴

Answer

(x - y)⁴

Answer

x⁴ - y⁴ is prime

Question 36

Which of the following expressions is a factor of the trinomial x² - 5x + 6?

Accepted Answer

(x - 2)

Answer

(x + 2)

Answer

(x - 6)

Answer

(x + 3)

Question 37

What is the factored form of the polynomial 2x² + 7x + 3?

Accepted Answer

(2x + 1)(x + 3)

Answer

(2x + 3)(x + 1)

Answer

(x + 1)(x + 3)

Answer

(2x - 1)(x - 3)

Question 38

Factor the polynomial 3x³ - 12x² + 9x completely.

Accepted Answer

3x(x - 1)(x - 3)

Answer

(3x² - 3x)(x - 3)

Answer

x(3x - 3)(x - 3)

Answer

3x(x² - 4x + 3)

Question 39

Which factoring method is most appropriate for the polynomial x² + 7x + 12?

Accepted Answer

Trial and Error

Answer

Difference of Squares

Answer

Sum or Difference of Cubes

Answer

Factoring by Grouping

Question 40

Factor the expression x⁴ - y⁴ completely.

Accepted Answer

(x² + y²)(x + y)(x - y)

Answer

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

Answer

(x^2 - y^2)^2

Answer

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

Question 41

Which factoring pattern can be used to factor the polynomial x³ + 8?

Accepted Answer

Sum of Cubes

Answer

Perfect Square Trinomial

Answer

Difference of Squares

Answer

Factoring by Grouping

Question 42

What are the solutions to the quadratic equation $x^2 - 4x - 5 = 0$ ?

Accepted Answer

x = 5, x = -1

Answer

x = 4, x = -1

Answer

x = -5, x = 1

Answer

x = -4, x = 1

Question 43

The area of a rectangular garden is represented by the polynomial $x^2 + 10x + 24$. If the length of the garden is (x + 6), what is the width?

Accepted Answer

(x + 4)

Answer

(x + 12)

Answer

(x + 2)

Answer

(x + 6)

Question 44

For what value of 'k' will the expression $x^2 + kx + 16$ be a perfect square trinomial?

Accepted Answer

k = 8 or k = -8

Answer

k = 4

Answer

k = -4

Answer

k = 16

Question 45

Which of the following polynomials is prime (not factorable over the integers)?

Accepted Answer

x^2 + x + 1

Answer

x^2 - 1

Answer

x^2 - 2x + 1

Answer

x^2 - 4