Question

what is the factored form of the polynomial? ( x^2 - 16x + 48 ) ( circ (x - 4)(x - 12) ) ( circ (x - 6)(x - 8) ) ( circ (x + 4)(x + 12) ) ( circ (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). The numbers will be negative because their product is positive and sum is negative.

Step2: Find the two numbers

We find that \(-4\) and \(-12\) multiply to \(48\) (\((-4)\times(-12) = 48\)) and add up to \(-16\) (\(-4 + (-12)=-16\)).

Step3: Write the factored form

So, \(x^2 - 16x + 48=(x - 4)(x - 12)\). We can also check by expanding the options:

• Expanding \((x - 4)(x - 12)\): \(x^2-12x - 4x + 48=x^2 - 16x + 48\), which matches.
• Expanding \((x - 6)(x - 8)\): \(x^2-8x - 6x + 48=x^2 - 14x + 48\), does not match.
• Expanding \((x + 4)(x + 12)\): \(x^2+12x + 4x + 48=x^2 + 16x + 48\), does not match.
• Expanding \((x + 6)(x + 8)\): \(x^2+8x + 6x + 48=x^2 + 14x + 48\), does not match.

Answer:

A. \((x - 4)(x - 12)\)