Problem 5 - A Graphical Approach to College Algebra

Question

Factor the greatest common factor from each polynomial. $$2(a+b)+4 m(a+b)$$

Step-by-Step Solution

Answer

The polynomial factors to \(2(a+b)(1 + 2m)\).

Step 1: Identify the Common Factors

Look at each term within the polynomial to find terms with common factors. We have two terms here: \(2(a+b)\) and \(4m(a+b)\). Notice that both terms include \((a+b)\) as a factor.

Step 2: Evaluate Numerical Coefficients

Evaluate the coefficients 2 and 4m to see if there is a common numerical factor. The greatest common factor (GCF) of these coefficients is 2, because 2 divides both 2 and 4.

Step 3: Factor Out the Greatest Common Factor

The GCF for the entire expression is \(2(a+b)\). Therefore, factor \(2(a+b)\) out of each term: \[2(a+b)(1 + 2m)\].

Step 4: Verify the Factorization

Multiply the factored expression to ensure it equals the original polynomial: \(2(a+b) \times (1 + 2m) = 2(a+b) + 4m(a+b)\). Verification shows that factoring is correct.

Key Concepts

Greatest Common Factor

The concept of the Greatest Common Factor (GCF) is crucial in simplifying algebraic expressions. It helps to identify the largest factor that divides two or more numbers or terms evenly. In polynomial factorization, finding the GCF is often the first step. Given terms like \(2(a+b)\) and \(4m(a+b)\), identifying \((a+b)\) as a common factor becomes straightforward. By examining the coefficients 2 and 4m, we determine that the numerical GCF is 2, which is the largest number that divides both coefficients evenly. Understanding the GCF not only simplifies calculations but also aids in breaking down complex polynomials into manageable parts. This simplification process is highly beneficial for further algebraic manipulations.

Factoring Polynomials

Factoring polynomials involves breaking down an expression into products of simpler expressions. This process is like reverse distribution: you take a polynomial and express it as a product of its factors. For our example, identifying that both terms \(2(a+b)\) and \(4m(a+b)\) include \((a+b)\) allows us to factor it out. After removing the repeated factor \((a+b)\), combine the remaining coefficients, giving us \(1 + 2m\) as the other factor. Thus, our factorized expression becomes \[2(a+b)(1+2m)\]. This not only simplifies the expression but also makes it easier to handle in subsequent steps like solving equations or performing integrations. Understanding the structure of polynomials through factorization is indispensable for learning higher-level mathematics.

Mathematical Verification

Verification is a crucial step in mathematics, providing assurance that a solution is correct. After factoring a polynomial, it's essential to check whether the newly formed expression, when multiplied out, returns to the original polynomial. In our case, multiplying the factors \(2(a+b)\) and \((1 + 2m)\) gives us \(2(a+b) + 4m(a+b)\). This step confirms the accuracy of our factorization. Verification ensures that any mistakes made during factoring are caught and corrected. This habit strengthens mathematical reasoning and boosts confidence in handling more complex problems. Moreover, verifying solutions fosters attention to detail, which is vital in mathematical problem-solving and broader analytical tasks.