Question

what is the factored form of the polynomial?
x² - 16x + 48
○ (x - 4)(x - 12)
○ (x - 6)(x - 8)
○ (x + 4)(x + 12)
○ (x + 6)(x + 8)

Explanation:

Step1: Recall factoring trinomials

To factor \(x^2 - 16x + 48\), we need two numbers that multiply to \(48\) and add up to \(-16\) (since the middle term is \(-16x\) and the constant term is positive, both numbers are negative).

Step2: Find the two numbers

We list the factor pairs of \(48\): \(1\) and \(48\), \(2\) and \(24\), \(3\) and \(16\), \(4\) and \(12\), \(6\) and \(8\). Among these, \(-4\) and \(-12\) multiply to \(48\) (\((-4)\times(-12) = 48\)) and add up to \(-16\) (\(-4 + (-12)=-16\)).

Step3: Write the factored form

Using these numbers, the factored form of \(x^2 - 16x + 48\) is \((x - 4)(x - 12)\). We can check by expanding: \((x - 4)(x - 12)=x^2-12x - 4x + 48=x^2 - 16x + 48\), which matches the original polynomial.

Answer:

\((x - 4)(x - 12)\) (corresponding to the first option: \((x - 4)(x - 12)\))