Problem 41 - Elementary Algebra

Question

Factor the greatest common factor from each polynomial. \(5 x(x+1)+3(x+1)\)

Step-by-Step Solution

Answer

The factored form is \( (x+1)(5x + 3) \)

Step 1: Identify Common Factor

Observe the polynomial and identify the common factor in each term. Here, both terms contain \(x+1\).

Step 2: Factor Out the Common Term

Factor \(x+1\) out of each term: \(5x(x+1) + 3(x+1) = (x+1)(5x + 3)\).

Key Concepts

Greatest Common Factor

Understanding the Greatest Common Factor (GCF) in polynomials is crucial. The GCF is the largest term that divides each term of the polynomial without leaving a remainder.
In our example, the polynomial is: \[5 x(x+1) + 3(x+1)\]
To find the GCF:

• Identify common factors in the polynomial's terms.
• Here, each term includes \(x+1\).
• Thus, our GCF is \(x+1\).

Factoring out the GCF simplifies the polynomial, making it easier to understand and solve.

Factoring Techniques

There are several factoring techniques to consider. These include:

• Factoring out the GCF: As shown in the example, we factor out \(x+1\). \[5x(x+1) + 3(x+1) = (x+1)(5x + 3)\]
• Factoring by grouping: Invite grouping terms and then factoring out the GCF.

Using these techniques will help break down complex polynomials into simpler, more manageable parts.

Polynomials

Polynomials are expressions with multiple terms, often separated by plus or minus signs. Each term consists of a coefficient (a number) and variables (like x or y) raised to exponents.
In our example, the polynomial is: \[5 x(x+1) + 3(x+1)\]
Recognizing the structure of polynomials is key to factoring them correctly.

• Observe each term's coefficients and variables.
• Identify common elements (terms, factors) to simplify the polynomial.

This understanding forms the foundation of successful polynomial factoring.