The polynomial x^2 + 8x + 15 can be factored into the form (x + a)(x + b), where a and b a… — ACT

The polynomial $x^2 + 8x + 15$ can be factored into the form $(x + a)(x + b)$ , where $a$ and $b$ are integers such that $a < b$ . What is the value of $2a + b$ ?

Answer: 11

Answer

The value of

2a + b

is 11.

Factoring the trinomial

x^2 + 8x + 15

gives

(x + 3)(x + 5)

. Since

a < b

, we must have

a = 3

and

b = 5

. Thus,

2a + b = 2(3) + 5 = 11

Step-by-Step Solution

Factor the quadratic expression

x^2 + 8x + 15

(x + 3)(x + 5)

To factor the trinomial, we find two integers that multiply to the constant term 15 and add to the linear coefficient 8. The integers 3 and 5 satisfy these requirements.

Assign the values to

a

and

b

under the condition

a < b

a = 3

and

b = 5

Comparing

(x + 3)(x + 5)

(x + a)(x + b)

gives the values 3 and 5. The condition

a < b

dictates that the smaller value 3 goes to

a

and the larger value 5 goes to

b

Calculate the value of

2a + b

Substitute

a = 3

and

b = 5

into the expression:

2(3) + 5 = 6 + 5 = 11

Key Concept

Factoring quadratic trinomials

Question

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