Question

the area of a rectangle is $(x^{3}-5x^{2}+3x - 15)$, and the width of the rectangle is $(x^{2}+3)$. if area = length × width, what is the length of the rectangle?
$x + 5$
$x-15$
$x + 15$
$x-5$

Explanation:

Step1: Factor the area expression

Factor \(x^{3}-5x^{2}+3x - 15\) by grouping.
Group as \((x^{3}-5x^{2})+(3x - 15)\).
Factor out \(x^{2}\) from the first group and \(3\) from the second group: \(x^{2}(x - 5)+3(x - 5)=(x^{2}+3)(x - 5)\).

Step2: Use the area formula

Since area \(=\text{length}\times\text{width}\), and area \(=(x^{2}+3)(x - 5)\), width \(=x^{2}+3\).
Then length \(=\frac{\text{area}}{\text{width}}=\frac{(x^{2}+3)(x - 5)}{x^{2}+3}\).
Cancel out the common factor \(x^{2}+3\).

Answer:

\(x - 5\) (the fourth option)