Question

select the correct answer.

factor the polynomial below.

\\x^2 - 16x + 63\\

• \\((x - 9)(x + 7)\\)
• \\((x - 9)(x - 7)\\)
• \\((x + 9)(x + 7)\\)
• \\((x + 9)(x - 7)\\)

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

We want to factor the quadratic expression:

$$ x^2 - 16x + 63 $$

This is in the standard form \( ax^2 + bx + c \), where:

• \( a = 1 \)
• \( b = -16 \)
• \( c = 63 \)

Step 2: Find two numbers that meet the conditions

We need to find two numbers that:

1. Multiply to give \( c = 63 \)
2. Add up to give \( b = -16 \)

Since the product is positive (\( +63 \)) and the sum is negative (\( -16 \)), both numbers must be negative.

Let's look at the factors of \( 63 \):

• \( -1 \times -63 = 63 \) (sum is \( -64 \))
• \( -3 \times -21 = 63 \) (sum is \( -24 \))
• \( -7 \times -9 = 63 \) (sum is \( -16 \))

The two numbers are \( -9 \) and \( -7 \).

Step 3: Write the factored form

Using the two numbers we found, we can write the expression as a product of two binomials:

$$ (x - 9)(x - 7) $$

Answer:

\( (x - 9)(x - 7) \)