91Ó°ÊÓ

Polynomial factorization is a method used to express a polynomial as a product of simpler polynomials. This makes the polynomial easier to solve or simplify. To factorize a polynomial, we look for patterns or techniques that allow us to break it down into simpler expressions.
In the given problem, we need to factor the polynomial \( n^2 b + 3 b + 15 + 5 n^2 \). By factorizing, we aim to express this as a product of two or more polynomials.

The Greatest Common Factor (GCF) is the highest expression that divides the terms of a polynomial without a remainder. Finding the GCF is crucial in factorization because it simplifies the polynomial, making further factorization easier.
To find the GCF, we identify the highest common factor in each group of terms. In our example, we have the polynomial \( n^2 b + 5 n^2 + 3 b + 15 \). We can group the terms into \( (n^2 b + 5n^2) \) and \( (3b + 15) \).
From the first group \( n^2 b + 5n^2 \), the GCF is \( n^2 \). We factor it out: \( n^2 (b + 5) \).
From the second group \( 3b + 15 \), the GCF is \( 3 \). We factor it out: \( 3 (b + 5) \).

The grouping method is a technique used to factor polynomials by grouping terms that share a common factor. It involves reorganizing and combining the polynomial's terms in pairs or groups, then factoring out the GCF from each group.
In our example, after rearranging the terms to \( n^2 b + 5 n^2 + 3 b + 15 \), we group terms into two pairs: \( (n^2 b + 5 n^2) \) and \( (3b + 15) \).
Factoring out the GCF from each group, we get: \( n^2 (b + 5) \) and \( 3 (b + 5) \).
The expression now is: \( n^2 (b + 5) + 3 (b + 5) \).
Since both groups contain the common factor \( (b + 5) \), we factor this out: \( (n^2 + 3)(b + 5) \).
The polynomial is now factored by grouping into \( (n^2 + 3)(b + 5) \).