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 \\(\times\\) width, what is the length of the rectangle?

\\(x + 5\\)
\\(x - 15\\)
\\(x + 15\\)
\\(x - 5\\)

Explanation:

Set up the relationship

Using the Polynomial Factoring and Polynomial Division knowledge points

$$ \text{Area} = \text{Length} \times \text{Width} $$

$$ x^3 - 5x^2 + 3x - 15 = \text{Length} \times (x^2 + 3) $$

$$ \text{Length} = \frac{x^3 - 5x^2 + 3x - 15}{x^2 + 3} $$

Factor the numerator

Using the Polynomial Factoring knowledge point

$$
LATEXBLOCK0
$$

Simplify the expression

Using the Polynomial Division knowledge point

$$ \text{Length} = \frac{(x^2 + 3)(x - 5)}{x^2 + 3} = x - 5 $$

Answer:

• \(x + 5\)
• \(x - 15\)
• \(x + 15\)
• \(x - 5\) (Correct answer)