Question

the area of a rectangle is (x³ - 5x² + 3x - 15), and the width of the rectangle is (x² + 3). if area = length × width, what is the length of the rectangle?
○ x + 5
○ x - 15
○ x + 15
○ x - 5

Explanation:

Step1: Use area formula to find length

Since area = length × width, then length = $\frac{\text{area}}{\text{width}}$. Given area = $x^{3}-5x^{2}+3x - 15$ and width = $x^{2}+3$, so length = $\frac{x^{3}-5x^{2}+3x - 15}{x^{2}+3}$.

Step2: Factor the numerator

Factor $x^{3}-5x^{2}+3x - 15$ by grouping. Group the terms: $(x^{3}-5x^{2})+(3x - 15)$. Factor out common factors from each group: $x^{2}(x - 5)+3(x - 5)=(x^{2}+3)(x - 5)$.

Step3: Simplify the fraction

Substitute the factored - form of the numerator into the length formula: $\frac{(x^{2}+3)(x - 5)}{x^{2}+3}$. Cancel out the common factor $(x^{2}+3)$ in the numerator and denominator. The result is $x - 5$.

Answer:

D. $x - 5$