91Ó°ÊÓ

Factoring algebraic expressions is an important aspect of solving many algebra problems. In this exercise, we first factor the quadratic expression \(x^2 + 6x + 9\). Notice that it resembles a perfect square trinomial, taking the form \((a + b)^2 = a^2 + 2ab + b^2\). Given our expression, we identify that:

• \(a^2 = x^2\)
• \(2ab = 6x\)
• \(b^2 = 9\)

We see that \(a = x\) and \(b = 3\), so \(x^2 + 6x + 9\) factors to \((x + 3)^2\). Recognizing patterns in expressions can make factoring easier.
Another part of the expression involves \(x^3 + 27\). Initially, it might not be clear how to factor it, but recognize this is a sum of cubes, which leads us into our next section on sum of cubes formula.

Polynomial division is a method that helps simplify expressions involving the division of one polynomial by another. In our problem, we simplify the expression \(\frac{(x+3)^2}{x^3 + 3^3} \cdot \frac{1}{x+3}\) through correct factoring and division.
Once the expressions \((x+3)^2\) and \(x^3 + 27\) are rewritten through factoring, we proceed to treat each part of the expression as individual factors that we can simplify.

• In the numerator, we have \((x+3)^2\), which can be expanded as \((x+3) \cdot (x+3)\).
• In the denominator, after applying the sum of cubes formula, we get \((x+3)(x^2-3x+9)\).

The \((x+3)\) in both the numerator and the denominator cancels out, a common occurrence in polynomial division called cancelation of common factors, leaving us with a simpler expression of \((x+3)/(x^2 - 3x + 9)\). Simplifying through polynomial division often requires recognizing such opportunities to cancel terms.

The sum of cubes formula is a handy tool for factoring expressions of the form \(a^3 + b^3\). In our exercise, \(x^3 + 27\) is a sum of cubes as it can be rewritten with \(a = x\) and \(b = 3\), since \(27 = 3^3\).

• The formula is \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\).

Apply it:

• \((x+3)\) is the \(a+b\) part.
• \(x^2 - 3x + 9\) is the \(a^2 - ab + b^2\) part.

Both these factors help simplify and cancel terms in the original mathematical expression. Using this formula allows us to break down seemingly complex polynomial expressions into simpler, factorable components, a key skill in algebra.