Factoring Polynomials

42 questions

All types42 Multiple choice22 Matching7 Fill in the blank8 Short answer5

Which of the following represents the completely factored form of the quadratic expression $3x^2 - 3x - 18$ ?

3(x - 3)(x + 2)

3(x + 3)(x - 2)

(3x - 9)(x + 2)

3(x - 6)(x + 1)

Show answer & explanation

Answer: $3(x - 3)(x + 2)$

Answer

The completely factored form is

3(x - 3)(x + 2)

The correct answer properly extracts the GCF of 3 first, leaving the trinomial

x^2 - x - 6

. This trinomial factors into

(x - 3)(x + 2)

because

-3

and

2

multiply to

-6

and add up to

-1

Step-by-Step Solution

Identify and factor out the Greatest Common Factor (GCF) from all terms in the expression.

3(x^2 - x - 6)

The coefficients

3, -3,

and

-18

are all divisible by

3

Find two numbers that multiply to the constant term (

-6

) and add to the coefficient of the middle term (

-1

The numbers are

-3

and

2

-3 \times 2 = -6

and

-3 + 2 = -1

Rewrite the quadratic trinomial as the product of two binomials using these numbers.

3(x - 3)(x + 2)

This represents the final, completely factored form of the original expression.

Key Concept

Factoring out a Greatest Common Factor (GCF) followed by factoring a quadratic trinomial.

Question 42Question →

Pair the given polynomial expressions on the left with their fully factored equivalents on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings are:

4x^2 - 25

with

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

;

x^2 - 8x + 15

with

(x - 3)(x - 5)

;

3x^2 + 12x - 15

with

3(x + 5)(x - 1)

; and

2x^3 - 8x

with

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

Each polynomial expression was broken down into its fundamental algebraic components by identifying patterns such as the difference of squares, standard trinomial factoring, and extracting the greatest common factor.

Step-by-Step Solution

Analyze the first expression

4x^2 - 25

It factors to

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

It follows the difference of squares pattern

a^2 - b^2 = (a - b)(a + b)

with

a = 2x

and

b = 5

Analyze the second expression

x^2 - 8x + 15

It factors to

(x - 3)(x - 5)

The constant term is positive and the middle coefficient is negative, so both factors must contain negative constants that multiply to

15

and add to

-8

Analyze the third expression

3x^2 + 12x - 15

It factors to

3(x + 5)(x - 1)

Factoring out the GCF of

3

gives

3(x^2 + 4x - 5)

. The resulting quadratic factors into binomials with constants

5

and

-1

Analyze the fourth expression

2x^3 - 8x

It factors to

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

Factoring out the GCF of

2x

gives

2x(x^2 - 4)

. The expression

x^2 - 4

is a difference of squares that further factors into

(x - 2)(x + 2)

Key Concept

Factoring Polynomials

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Factoring Polynomials

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes