Determine the missing constants in the completely factored form of the given polynomial.

Answer:The polynomial

5x^2 - 5x - 60

can be factored completely. When written in the form

a(x - b)(x + c)

, where

b

and

c

are positive integers, the value of

a

is 【5】, the value of

b

is 【4】, and the value of

c

is 【3】.

Answer

The constants are a = 5, b = 4, and c = 3.

The correct values are found by first factoring out the greatest common factor of

5

, leaving the quadratic expression

x^2 - x - 12

. This quadratic expression factors into two binomials by finding two numbers that multiply to

-12

and add to

-1

, which are

-4

and

3

. This yields the fully factored form

5(x - 4)(x + 3)

. By matching this result to the required algebraic structure

a(x - b)(x + c)

, we can directly identify that

a = 5

b = 4

, and

c = 3

Step-by-Step Solution

Find the greatest common factor (GCF) of the terms in the polynomial

5x^2 - 5x - 60

The GCF is

5

. Factoring it out gives

5(x^2 - x - 12)

Extracting the GCF simplifies the trinomial and makes it easier to factor into binomials.

Factor the remaining quadratic trinomial

x^2 - x - 12

We look for two integers that multiply to

-12

and add to

-1

This is the standard procedure for factoring a monic quadratic trinomial of the form

x^2 + bx + c

Identify the proper integers and write the binomial factors.

The integers are

-4

and

3

. The factored trinomial is

(x - 4)(x + 3)

(-4) \times 3 = -12

and

-4 + 3 = -1

Combine the factors and match them to the target form

a(x - b)(x + c)

The completely factored expression is

5(x - 4)(x + 3)

. Matching this to

a(x - b)(x + c)

yields

a = 5

b = 4

, and

c = 3

This establishes the precise values corresponding to the blanks in the question.

Key Concept

Factoring out a greatest common factor followed by factoring a quadratic trinomial.

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