The rectangle below has an area of \( x^{2}+13 x+36 \) square meters and a length of \( x+9 \) meters. What expression represents the width of the rectangle?

To find the width of the rectangle, we can use the area formula for a rectangle, which is Area = Length × Width. Given that the area is \( x^{2}+13 x+36 \) square meters and the length is \( x+9 \) meters, we can set up the equation: \( x^{2}+13 x+36 = (x+9) \cdot \text{Width} \). To solve for Width, we can divide the area by the length: \[ \text{Width} = \frac{x^{2}+13x+36}{x+9}. \] Using polynomial long division or synthetic division, we find that \( x^{2}+13x+36 \) can be factored as \( (x+9)(x+4) \). Hence, \[ \text{Width} = x + 4. \] The expression that represents the width of the rectangle is \( x + 4 \) meters. Now, about polynomial division! It’s a handy tool that can simplify expressions, and it's like breaking down a puzzle piece by piece. So, whether you're dealing with quadratics or higher polynomials, knowing how to divide them can save you time and effort on tougher math problems! If you're interested in diving deeper into quadratic equations, exploring factoring techniques and the quadratic formula can be rewarding. Not only do these concepts deepen your understanding of algebra, but they also have connections to areas like physics and engineering, where parabolic trajectories are a key component of projectile motion!