Question

factor the polynomial completely using the x method.

\\(x^2 + 16x + 48\\)

which equivalent four-term polynomial can be created using the x method?

\\(x^2 + 8x + 8x + 48\\)
\\(x^2 - 12x - 4x + 48\\)
\\(x^2 + 12x + 4x + 48\\)
\\(x^2 - 8x - 8x + 48\\)

Explanation:

Identify coefficients for the X method

For the trinomial \(x^2 + 16x + 48\), the standard form is \(ax^2 + bx + c\):

$$ a = 1, \quad b = 16, \quad c = 48 $$

The top of the X contains \(ac = 1 \cdot 48 = 48\), and the bottom contains \(b = 16\).

Find two factors

We need two numbers that multiply to \(ac = 48\) and add to \(b = 16\):

$$ 12 \cdot 4 = 48 \quad \text{and} \quad 12 + 4 = 16 $$

Rewrite the middle term

Using the factors \(12\) and \(4\), split the middle term \(16x\) into \(12x + 4x\):

$$ x^2 + 12x + 4x + 48 $$

Answer:

• (A) \(x^2 + 8x + 8x + 48\)
• (B) \(x^2 - 12x - 4x + 48\)
• (C) \(x^2 + 12x + 4x + 48\) (Correct answer)
• (D) \(x^2 - 8x - 8x + 48\)