The polynomial x^2 + 6x - 16 can be factored as (x - 2)(x + k). What is the value of the c… — Armed Services Vocational Aptitude Battery (ASVAB)

The polynomial $x^2 + 6x - 16$ can be factored as $(x - 2)(x + k)$ . What is the value of the constant $k$ ?

Answer: 8 / +8 / eight

The constant

k

must be 8 because in the factored form

(x - 2)(x + k)

, the product of -2 and

k

must equal the constant term of the polynomial, which is -16. Since

-2 \times 8 = -16

and the sum

-2 + 8 = 6

gives the correct middle coefficient, the value of

k

is exactly 8.

Analyze the relationship between the expanded polynomial and its factored form.

The expanded form is

x^2 + 6x - 16

and the factored form is

(x - 2)(x + k)

This sets up the equation to solve for the missing constant

k

Use the property of multiplying binomials to find the constant term.

The product of the constants in the binomials must equal the constant term of the polynomial:

-2 \times k = -16

When expanding

(x + a)(x + b)

, the constant term is always equal to

a \times b

Solve the equation for

k

k = -16 \div (-2) = 8

Dividing both sides by -2 isolates

k

Verify the middle term coefficient with the found value of

k

-2 + 8 = 6

, which matches the middle term

6x

Checking the sum of the constants ensures the entire factorization is correct.

Factoring quadratic trinomials by finding two numbers that multiply to the constant term and add to the middle coefficient.

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