91Ó°ÊÓ

Factoring polynomials is essential for simplifying rational expressions. When you factor a polynomial, you are rewriting it as a product of simpler polynomials. For example, take the expression \(4p + 12\). Notice that both terms share a common factor of 4. We can factor this out, resulting in \(4(p + 3)\). This step simplifies working with rational expressions.

Similarly, consider \(5p - 5\). Here, both terms share a common factor of 5. Factoring out the 5, you get \(5(p - 1)\). This makes the expression easier to manage.

Key points to remember:

• Identify the greatest common factor (GCF) of the terms.
• Factor out the GCF from the polynomial.
• Rewrite the polynomial as a product of the GCF and the remaining terms.

Multiplying rational expressions involves a few key steps. First, you need to factor both the numerators and denominators. This ensures that you can simplify the expressions before multiplying them together.

For example, suppose you have two rational expressions: \(\frac{5(p - 1)}{4(p + 3)}\) and \(\frac{10(p + 1)}{7(p + 3)}\).

Once you've factored them, you rewrite the division of these two rational expressions as a multiplication involving the reciprocal of the second rational expression:
\[ \frac{5(p - 1)}{4(p + 3)}\ \times \ \frac{7(p + 3)}{10(p + 1)}\]

Now, you can multiply the numerators and the denominators as follows:

• Multiply the numerators: \(5(p - 1) \times 7(p + 3)\)
• Multiply the denominators: \(4(p + 3) \times 10(p + 1)\)

This multiplication simplifies to:
\[\frac{35(p - 1)}{40(p + 1)}\]
by reducing the common factors.

To further simplify rational expressions, you often need to cancel common factors in the numerator and the denominator. This step ensures that the expression is in its simplest form.

Consider our example: \(\frac{5(p - 1)}{4(p + 3)} \times \ \frac{7(p + 3)}{10(p + 1)}\). Before multiplying, you will notice that \(p + 3\) appears in both the numerator and the denominator.

You can cancel these common factors, leading to a simplified version of the expression:
\[\frac{5(p - 1)}{4} \times \ \frac{7}{10(p + 1)}\]

• Simplify by multiplying the remaining factors.
• Simplify the expression further if possible.

Following these steps results in: \[\frac{35(p - 1)}{40(p + 1)}\].
Canceling common factors is crucial for expressing an answer clearly and succinctly.