Problem 20 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. $$20 m-5 m^{2}$$

Step-by-Step Solution

Answer

The short answer to the problem is: \(5m(4 - m)\).

Step 1: Identify the greatest common factor (GCF)

To find the GCF of \(20m\) and \(-5m^2\), we will first find the GCF of the coefficients (20 and -5) and then find the GCF of the variable part (m and m^2). The GCF of 20 and -5 is 5 because 5 is the largest factor that divides both numbers. The GCF of m and \(m^2\) is m since it is the smallest power of m that divides both terms. Therefore, the GCF of the given terms is 5m.

Step 2: Factor out the greatest common factor

Now that we have identified the GCF as 5m, we can factor it out of each term. \(20m - 5m^2 = 5m(4 - m)\) In this step, we have factored out 5m from both terms of the given expression.

Step 3: Check the answer

To make sure our factored expression is correct, we will expand the factored form and verify that it matches the original expression. \((5m)(4 - m) = 5m * 4 - 5m * m = 20m - 5m^2\) Our factored expression is correct since it matches the original expression. Therefore, the factored form of the given expression is \(\boxed{5m(4 - m)}\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest term that divides all the terms of an expression without leaving a remainder. In the context of algebraic expressions, it is often found by determining the greatest divisor of both the numerical coefficients and the variables.

To find the GCF, follow these steps:

• Numerical Coefficients: Analyze each term's coefficients separately, identifying the largest number that divides each. For instance, in the problem where we consider the coefficients 20 and -5, the GCF is 5.
• Variables: For variables, use the smallest power of the variable that is present in all terms. In our example, between \(m\) and \(m^2\), the variable GCF is \(m\).

Finding the correct GCF plays a crucial role in simplifying expressions, easing further algebraic manipulations, and ultimately leads to accurate solutions.

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial into simpler "factor" polynomials that, when multiplied together, give the original polynomial. It is a critical process in algebra since it transforms complex expressions into more manageable forms.

To factor a polynomial:

• Firstly, determine if there is a common factor among the terms of the polynomial, just as demonstrated in the GCF section.
• Divide each term of the polynomial by the identified GCF. For example, in the expression \(20m - 5m^2\), once we've determined that the GCF is \(5m\), we divide to simplify the remaining terms:
• The expression \(20m - 5m^2\) simplifies to \(5m(4 - m)\) after factoring out \(5m\).

Employing polynomial factoring not only simplifies expressions but enables solving equations, sketching graphs, and understanding polynomial functions more profoundly.

Algebraic Techniques

Algebraic techniques refer to various methods used to manipulate expressions and solve equations. Mastering these techniques is essential for solving real-world problems effectively. Factoring, using the distributive law, and recognizing patterns are among the core techniques you'll use regularly.

Key algebraic techniques include:

• Distributive Property: This property allows you to multiply a single term by each of the terms inside a bracket, which you often use to check factored expressions. For instance, expanding \((5m)(4 - m)\) confirms it equals our original polynomial \(20m - 5m^2\).
• Expression Simplification: Through operations like factoring, simplify complex algebraic expressions and make them easier to solve. This includes the use of GCF and various factorization models.
• Pattern Recognition: Identify common patterns or structures within polynomials that might indicate how to factor them. Recognizing these patterns aids quicker and more efficient problem-solving.

Adopting these techniques enables seamless navigation through even the most challenging algebraic tasks and fosters a deeper understanding of mathematical theory.