Problem 2 - Elementary Algebra

Question

Use the grouping method to factor the following polynomials. $$ 2 a m+8 m+5 a n+20 n $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial is \((a + 4)(2m + 5n)\).

Step 1: Group the terms with common factors

First, let's group the terms with common factors. In this case, it is quite clear that the terms with 'm' have a common factor and the terms with 'n' have a common factor. So, we can write the given polynomial as follows: $$ (2am + 8m) + (5an + 20n) $$

Step 2: Factor out the common factors in each group

Now, let's factor out the common factors in each group: For the first group \((2am + 8m)\), the common factor is '2m'. When we factor out '2m', we get: $$ 2m(a + 4) $$ For the second group \((5an + 20n)\), the common factor is '5n'. Factoring out '5n', we have: $$ 5n(a + 4) $$

Step 3: Write the factored groups in the polynomial

Now let's write the polynomial with the factored groups: $$ 2m(a + 4) + 5n(a + 4) $$

Step 4: Factor out the common binomial factor

In this final step, we can see that both terms have a common binomial factor of \((a + 4)\). We can factor this out to get the final factored form: $$ (a + 4)(2m + 5n) $$ So, the given polynomial has been factored using the grouping method as: $$ (a + 4)(2m + 5n) $$

Key Concepts

Understanding Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, combined using only addition, subtraction, and multiplication operations. Importantly, the powers of the variables are non-negative integers. Polynomials can have one or more terms, and each term is made up of:

• A coefficient: A numerical factor that multiplies the variable(s).
• Variables: Symbols such as "x" or "y" that represent unknown quantities.
• Exponents: The powers to which the variables are raised.

A simple example of a polynomial is \(3x^2 + 5x - 7\). Here, each term is composed of a coefficient, variable, and exponent. The degree of a polynomial is determined by the highest exponent of its variable. Understanding polynomials is crucial in algebra, as they form the basis for operations such as factorization and solving equations.

The Grouping Method for Factorization

The grouping method is a useful technique in algebra for factoring polynomials that cannot be simplified using common factor methods directly. This method involves:

• Grouping terms within the polynomial that share common factors.
• Factoring out the common factors from each group.
• Rewriting the polynomial with these factored groups.
• Identifying and factoring out any common binomial factors that arise.

To employ the grouping method, you first identify pairs or groups of terms that have some shared elements that can be factored. Once this is done, you factor the polynomial into two or more binomial components. As shown in the example with \(2am+8m+5an+20n\), the terms were grouped and factored first by extracting the greatest common factors from each pair, leading to a new expression that could be further simplified by factoring a common binomial factor, resulting in \((a+4)(2m+5n)\). By mastering the grouping method, you gain a powerful tool for simplifying complex algebraic expressions.

Exploring Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. The flexibility of these expressions allows mathematicians to represent a wide variety of problems in a compact form.

• Variables in these expressions are placeholders for values we either don’t know or want to explore.
• Coefficients are numerical parts that multiply the variables.
• Operators are the symbols that denote mathematical operations.

In algebra, expressions can be manipulated by various operations, including simplification of terms, expansion, and especially factorization, similar to what we do with polynomials. One of the core objectives is to either solve the expression for variable values or transform it into a more usable form, such as factoring. For example, when we factor \(2am + 8m + 5an + 20n\), we are effectively breaking the expression down into products of simpler expressions, making it easier to work with. This process is vital for solving equations, graphing functions, and understanding relationships between quantities in algebra.