Question

the area of a rectangle is 4x² - 16. which binomial factor could represent the width of the rectangle? a x + 2 b x + 4 c 2x + 1 d 2x + 16

Explanation:

Step1: Factor the area expression

The area of the rectangle is \(4x^{2}-16\). Factor out the common factor \(4\) first: \(4(x^{2}-4)\).
Then, using the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\) (where \(a=x\) and \(b = 2\)), we get \(4(x + 2)(x - 2)\).

Step2: Analyze the binomial factors

If we assume one of the binomial factors (since the area of a rectangle \(A=\text{length}\times\text{width}\)) and considering the options:

• Option A: If the width is \(x + 2\), then the length could be \(4(x - 2)\) (because \(4(x + 2)(x - 2)=(x + 2)\times4(x - 2)\)).
• Option B: If we set \(x+4\) as a factor, \(4(x + 2)(x - 2)\div(x + 4)\) does not give a polynomial result.
• Option C: \(4(x + 2)(x - 2)\div(2x + 1)\) does not give a polynomial result.
• Option D: \(4(x + 2)(x - 2)\div(2x+16)=4(x + 2)(x - 2)\div2(x + 8)=2\frac{(x + 2)(x - 2)}{x + 8}\), which is not a polynomial result.

So the binomial factor that could represent the width of the rectangle is \(x + 2\).

Answer:

A. \(x + 2\)