91Ó°ÊÓ

Factoring polynomials involves breaking down a complex polynomial into simpler polynomials that multiply together to give the original polynomial. This is particularly useful when simplifying rational expressions.

For example, consider the polynomial in the numerator, \(10a^2 + 20a - 30\). By identifying the greatest common factor, we notice that \(10\) is a common factor. So, we can factor it out:
\[10a^2 + 20a - 30 = 10(a^2 + 2a - 3)\]
Breaking down polynomials helps in recognizing simpler forms and makes further operations, such as cancellations, easier.

The Greatest Common Factor (GCF) is the largest factor that two or more numbers share. In polynomials, the GCF is the highest polynomial that divides all terms of the polynomial without a remainder. Finding the GCF simplifies expressions and helps in further factoring.

For our problem, the GCF for the numerator \(10a^2 + 20a - 30\) is \(10\). Similarly, for the denominator \(5a^2 + 20a + 15\), the GCF is \(5\). We extract these GCFs first:
\[10a^2 + 20a - 30 = 10(a^2 + 2a - 3)\]
\[5a^2 + 20a + 15 = 5(a^2 + 4a + 3)\]
This step greatly simplifies the process of reducing the rational expression.

Quadratic expressions are polynomials where the highest exponent is two. These are often in the form \(ax^2 + bx + c\). When dealing with such expressions, factoring them can simplify operations like division.

In our solution, the quadratic expressions in the factored forms are \(a^2 + 2a - 3\) and \(a^2 + 4a + 3\). To factor these expressions, we look for two numbers that multiply to give the constant term \(c\) and add to give the coefficient \(b\). For the numerator, \(a^2 + 2a - 3\) can be factored as:
\[a^2 + 2a - 3 = (a + 3)(a - 1)\]
Similarly, for the denominator, \(a^2 + 4a + 3\), the factorization is:
\[a^2 + 4a + 3 = (a + 3)(a + 1)\]
Factoring quadratics into binomials helps in simplifying and reducing rational expressions.

Canceling common factors is a crucial step in simplifying rational expressions. After factoring polynomials in both the numerator and denominator, we can check for identical factors that appear in both parts.

In our simplified form, we have:
\[2 \frac{(a + 3)(a - 1)}{(a + 3)(a + 1)}\]
Here, \((a + 3)\) is a common factor in both the numerator and the denominator. By canceling the \((a + 3)\) terms, we get:
\[2 \frac{(a - 1)}{(a + 1)}\]
This step reduces the rational expression to its simplest form, making it easier to understand and work with further.