When the polynomial 5x^2 - 20 is factored completely into the form k(x - c)(x + c), where … — Armed Services Vocational Aptitude Battery (ASVAB)

When the polynomial $5x^2 - 20$ is factored completely into the form $k(x - c)(x + c)$ , where $k$ and $c$ are positive integers, what is the value of the sum $k + c$ ?

Answer: 7 / seven

Answer

The polynomial

5x^2 - 20

can be factored completely by first pulling out the greatest common factor,

5

, leaving

5(x^2 - 4)

. The expression inside the parentheses is a difference of squares and factors into

(x - 2)(x + 2)

. This makes the completely factored form

5(x - 2)(x + 2)

. Comparing this to the form

k(x - c)(x + c)

, we see that

k = 5

and

c = 2

. Their sum is

5 + 2 = 7

Step-by-Step Solution

Factor out the greatest common factor (GCF) from the polynomial.

5x^2 - 20 = 5(x^2 - 4)

Both terms share a common numerical factor of

5

. Factoring it out simplifies the remaining polynomial.

Factor the remaining binomial as a difference of squares.

5(x^2 - 4) = 5(x - 2)(x + 2)

The binomial

x^2 - 4

follows the pattern

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

, where

a = x

and

b = 2

Identify the values of

k

and

c

from the given form.

k = 5

and

c = 2

By matching

5(x - 2)(x + 2)

to the target form

k(x - c)(x + c)

, the constant multiplier is

5

and the positive integer in the binomial is

2

Calculate the sum of

k

and

c

5 + 2 = 7

The question asks for the final value of the sum

k + c

Key Concept

Factoring polynomials using the greatest common factor and the difference of squares.

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