Question

factor the trinomial completely.

\\x^2 + 15x + 54\\

select the correct choice below and, if necessary, fill in the answer box within your choice.

a. \\x^2 + 15x + 54 = \\
b. the polynomial is prime.

Explanation:

🆕 New Concept Discovered: Factoring Trinomials
Finding two numbers that multiply to the constant and add to the middle coefficient.

Step 1: Identify the coefficients

For the trinomial \( x^2 + 15x + 54 \), we identify the coefficients in the standard form \( ax^2 + bx + c \):

• \( a = 1 \)
• \( b = 15 \)
• \( c = 54 \)

We need to find two numbers that:

1. Multiply to \( c = 54 \)
2. Add up to \( b = 15 \)

Step 2: Find the factors of 54

Let's list the pairs of factors of \( 54 \) and find their sums:

• \( 1 \times 54 = 54 \) (Sum: \( 1 + 54 = 55 \))
• \( 2 \times 27 = 54 \) (Sum: \( 2 + 27 = 29 \))
• \( 3 \times 18 = 54 \) (Sum: \( 3 + 18 = 21 \))
• \( 6 \times 9 = 54 \) (Sum: \( 6 + 9 = 15 \))

The pair that multiplies to \( 54 \) and adds to \( 15 \) is \( 6 \) and \( 9 \).

Step 3: Write the factored form

Since the leading coefficient \( a = 1 \), we can write the factored form directly using our two numbers, \( 6 \) and \( 9 \):

$$ (x + 6)(x + 9) $$

Answer:

A. \( x^2 + 15x + 54 = (x + 6)(x + 9) \)