Question

1. write an expression to represent the area of the rectangle below in simplest form. (\frac{2x^3 - 32x}{x + 4}) rectangle (\frac{\frac{1}{x} - \frac{4}{x^2}}{4(x - 4)^2})

Explanation:

Step1: Recall the area formula for a rectangle

The area \( A \) of a rectangle is given by the product of its length \( l \) and width \( w \), i.e., \( A = l\times w \). Here, the length is \( \frac{2x^{3}-32x}{x + 4} \) and the width is \( \frac{\frac{1}{x}-\frac{4}{x^{2}}}{4(x - 4)^{2}} \). So we need to multiply these two rational expressions.

Step2: Simplify the numerator of the length expression

Factor the numerator of \( \frac{2x^{3}-32x}{x + 4} \). First, factor out the greatest common factor \( 2x \): \( 2x^{3}-32x=2x(x^{2}-16) \). Then, use the difference of squares formula \( a^{2}-b^{2}=(a + b)(a - b) \) where \( a=x \) and \( b = 4 \), so \( x^{2}-16=(x + 4)(x - 4) \). Thus, \( 2x^{3}-32x=2x(x + 4)(x - 4) \). So the length becomes \( \frac{2x(x + 4)(x - 4)}{x + 4} \). Cancel out the common factor \( x + 4 \) (assuming \( x
eq - 4 \)) to get \( 2x(x - 4) \).

Step3: Simplify the numerator of the width expression

Simplify the numerator \( \frac{1}{x}-\frac{4}{x^{2}} \). Find a common denominator, which is \( x^{2} \). So \( \frac{1}{x}-\frac{4}{x^{2}}=\frac{x}{x^{2}}-\frac{4}{x^{2}}=\frac{x - 4}{x^{2}} \). Thus, the width expression becomes \( \frac{\frac{x - 4}{x^{2}}}{4(x - 4)^{2}} \).

Step4: Simplify the width expression

To divide by a fraction, we multiply by its reciprocal. So \( \frac{\frac{x - 4}{x^{2}}}{4(x - 4)^{2}}=\frac{x - 4}{x^{2}}\times\frac{1}{4(x - 4)^{2}}=\frac{1}{4x^{2}(x - 4)} \) (assuming \( x
eq4 \) to cancel out \( x - 4 \)).

Step5: Multiply the simplified length and width

Now, multiply the simplified length \( 2x(x - 4) \) and the simplified width \( \frac{1}{4x^{2}(x - 4)} \). So \( 2x(x - 4)\times\frac{1}{4x^{2}(x - 4)} \). Cancel out the common factors: \( x \) cancels with one \( x \) in \( x^{2} \), \( (x - 4) \) cancels out, and \( 2 \) and \( 4 \) simplify to \( \frac{1}{2} \). So we have \( \frac{2x(x - 4)}{4x^{2}(x - 4)}=\frac{1}{2x} \).

Answer:

\(\frac{1}{2x}\)