Question

the area (in square inches) of a rectangle is given by the polynomial function \\(a(n) = n^2 + 12n + 27\\). if the length of the rectangle is \\((n + 3)\\) inches, what is the width?

Explanation:

Identify the given values and formula

We are given the area of a rectangle as a polynomial function:

$$ A(n) = n^2 + 12n + 27 $$

The length of the rectangle is:

$$ L(n) = n + 3 $$

Using the Geometric Polynomial Modeling concept, the area of a rectangle is the product of its length and width:

$$ A(n) = \text{length} \times \text{width} $$

Therefore, to find the width, we must divide the area by the length:

$$ W(n) = \frac{A(n)}{L(n)} = \frac{n^2 + 12n + 27}{n + 3} $$

Factor the quadratic expression

To simplify the division, we can factor the numerator.
Using the Factoring Trinomials concept, we look for two numbers that multiply to \(27\) and add up to \(12\).
These two numbers are \(9\) and \(3\), since:

$$ 9 \times 3 = 27 $$

$$ 9 + 3 = 12 $$

Thus, we can write the trinomial as:

$$ n^2 + 12n + 27 = (n + 9)(n + 3) $$

Simplify the rational expression

Now, substitute the factored form back into our width equation:

$$ W(n) = \frac{(n + 9)(n + 3)}{n + 3} $$

We can cancel the common factor of \((n + 3)\) from both the numerator and the denominator:

$$ W(n) = n + 9 $$

This gives us the width of the rectangle.

Answer:

The area (in square inches) of a rectangle is given by the polynomial function \(A(n) = n^2 + 12n + 27\). If the length of the rectangle is \((n + 3)\) inches, what is the width? <blank>\(n + 9\)</blank> inches