Question

1. geometry the area of a rectangle is ( x^3 + 8x^2 + 13x - 12 ) square units. the width of the rectangle is ( x + 4 ) units. what is the length of the rectangle?

Explanation:

Step1: Recall the area formula for a rectangle

The area \( A \) of a rectangle is given by \( A = \text{length} \times \text{width} \). So, to find the length, we can use the formula \( \text{length} = \frac{\text{area}}{\text{width}} \).

Step2: Identify the area and width

The area of the rectangle is \( x^{3}+8x^{2}+13x - 12 \) square units and the width is \( x + 4 \) units. So we need to perform polynomial long division or factor the cubic polynomial to divide by \( x + 4 \). Let's try polynomial long division.

Divide \( x^{3}+8x^{2}+13x - 12 \) by \( x + 4 \):

• First term: \( \frac{x^{3}}{x}=x^{2} \). Multiply \( x + 4 \) by \( x^{2} \) to get \( x^{3}+4x^{2} \). Subtract this from the dividend: \( (x^{3}+8x^{2}+13x - 12)-(x^{3}+4x^{2}) = 4x^{2}+13x - 12 \).
• Second term: \( \frac{4x^{2}}{x}=4x \). Multiply \( x + 4 \) by \( 4x \) to get \( 4x^{2}+16x \). Subtract this from the previous result: \( (4x^{2}+13x - 12)-(4x^{2}+16x)= - 3x - 12 \).
• Third term: \( \frac{-3x}{x}=-3 \). Multiply \( x + 4 \) by \( - 3 \) to get \( -3x - 12 \). Subtract this from the previous result: \( (-3x - 12)-(-3x - 12)=0 \).

So the quotient is \( x^{2}+4x - 3 \).

Answer:

The length of the rectangle is \( x^{2}+4x - 3 \) units.