In Exercises 45–68, factor by grouping.x² + 3x + 5x + 15

Verified step by step guidance

1. Group the terms into two pairs: (x^2 + 3x) and (5x + 15)
2. Factor out the greatest common factor from each pair: x(x + 3) + 5(x + 3)
3. Notice that (x + 3) is a common factor in both terms
4. Factor out the common factor (x + 3): (x + 3)(x + 5)
5. Verify the factorization by expanding: (x + 3)(x + 5) = x^2 + 5x + 3x + 15 = x^2 + 8x + 15

Key Concepts

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials with four or more terms. This technique involves rearranging the terms into two groups, factoring out the common factors from each group, and then factoring out the common binomial factor. It is particularly useful when the polynomial does not have a straightforward factorization.

Common Factors

A common factor is a number or variable that divides two or more terms without leaving a remainder. In the context of polynomials, identifying common factors within groups of terms is essential for simplifying expressions. Recognizing these factors allows for easier manipulation and ultimately leads to the correct factorization of the polynomial.

Binomial Factor

A binomial factor is a polynomial that consists of two terms, typically expressed in the form (ax + b). In the process of factoring by grouping, once common factors are extracted, the remaining expression often simplifies to a binomial. Recognizing and factoring out these binomials is crucial for completing the factorization of the original polynomial.