Complete the factored form of the given quadratic polynomial by determining the missing co… — Armed Services Vocational Aptitude Battery (ASVAB)

Complete the factored form of the given quadratic polynomial by determining the missing constant.

What number correctly fills in the blank?

Answer:

n^2 + 11n + 24 = (n + 3)(n + 【8】)

The polynomial

n^2 + 11n + 24

factors into two binomials. We are given one factor,

(n + 3)

. To find the missing value in the second factor

(n + b)

, we can use the property that the product of the constant terms in the binomials must equal the constant term of the trinomial. Therefore,

3 \times b = 24

, which means

b = 8

. We verify this by adding the constants:

3 + 8 = 11

, which matches the coefficient of the middle term. Thus, the correct value for the blank is 8.

Identify the relationship between the constant term of the polynomial and the constants in its binomial factors.

The product of the constants in the binomial factors must equal the constant term of the polynomial, 24.

When expanding

(n + a)(n + b)

, the constant term is calculated as

a \times b

Set up an equation to find the missing value.

3 \times \text{missing value} = 24

One of the binomial factors is given as

(n + 3)

, so one constant is 3.

Solve for the missing value and verify with the middle term.

The missing value is 8, since

24 \div 3 = 8

. Checking the middle term,

3 + 8 = 11

, which matches

11n

This confirms that 8 satisfies both the multiplication to 24 and the addition to 11 requirements for factoring the trinomial.

Factoring quadratic trinomials of the form x^2 + bx + c.

Question