91Ó°ÊÓ

Factoring polynomials means expressing a polynomial as a product of simpler polynomials. The goal is to break down a complicated expression into smaller, more manageable pieces. For example, when asked to factor a polynomial like the numerator in the given expression, we start by looking for a greatest common factor (GCF). In this case, the polynomial in the numerator is \(3x^2 - 27\).

Notice that both terms, \(3x^2\) and \(27\), share a common factor of 3. We can factor this out to get \(3(x^2 - 9)\). Now, we notice that \(x^2 - 9\) is also a special case known as a "difference of squares."

• The first term \(x^2\) is a square because it's \(x \times x\).
• The second term \(9\) is also a square since it's \(3 \times 3\).
• The expression is a difference of squares, so it can be factored further into \((x - 3)(x + 3)\).

Thus, the factoring of the numerator becomes \(3(x - 3)(x + 3)\). This type of insight simplifies algebraic expressions significantly.

The difference of squares is a special factoring technique used when both terms in a binomial are perfect squares and are separated by a subtraction sign. The general formula is \(a^2 - b^2 = (a - b)(a + b)\). This quick form of factoring can save time and simplify expressions considerably, like \(x^2 - 9\), where we see that both \(x^2\) and 9 are perfect squares.

Using the formula:

• Set \(a = x\), because \(x^2 = (x)^2\).
• Set \(b = 3\), because \(9 = (3)^2\).
• Then apply the formula: \((x - 3)(x + 3)\).

This strategy provides a straightforward way to break down what might initially seem to be complex expressions and can be handy for various algebraic operations, including simplifying expressions or solving equations.

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions such as polynomials. Simplifying these fractions involves a few key steps, primarily focusing on factoring. For instance, in the given exercise, after factoring both the numerator as \(3(x - 3)(x + 3)\) and the denominator as \((2x + 1)(x - 3)\), you can cancel out the common factors.

This process of canceling out is possible only when a term appears in both the numerator and the denominator. In this example, \((x - 3)\) is a common factor:

• Cancel \((x - 3)\) from both parts.
• You're left with \(\frac{3(x + 3)}{2x + 1}\).

After canceling out, always check if the remaining expression can be simplified further. However, in this case, there are no more common factors. Understanding how to handle algebraic fractions effectively helps simplify equations and aids in solving many types of algebraic problems.