Problem 40 - Elementary Algebra

Question

For the following problems, use the grouping method to factor the polynomials. Some may not be factorable. $$ a(2 x+7)-4(2 x+7)+a(x-10)-4(x-10) $$

Step-by-Step Solution

Answer

Question: Factor the given polynomial using the grouping method: a(2x + 7) - 4(2x + 7) + a(x - 10) - 4(x - 10). Answer: The factored form of the given polynomial using the grouping method is (x - 1)(3a - 12).

Step 1: Write down the given expression

We start by writing down the given polynomial expression: $$ a(2x + 7) - 4(2x + 7) + a(x - 10) - 4(x - 10) $$

Step 2: Distribute the constants to terms inside the parentheses

In this step, distribute the constants to the terms inside each parenthesis: $$ (2ax + 7a) - (8x + 28) + (ax - 10a) - (4x - 40) $$

Step 3: Rewrite the expression with similar terms grouped together

Now, rewrite the expression by grouping similar terms together: $$ (2ax + ax) + (-8x - 4x) + (7a - 10a) + (-28 + 40) $$

Step 4: Combine the similar terms

Next, combine the similar terms in each group: $$ (3ax) + (-12x) + (-3a) + (12) $$

Step 5: Factor by grouping

Now, factor by grouping the terms, finding the common factor in each group, and factoring it out: $$ 3ax - 3a - 12x + 12 $$ Group the expression into two pairs: $$ (3ax - 3a) + (-12x + 12) $$ Factor out the common factor from each pair: $$ 3a(x - 1) - 12(x - 1) $$

Step 6: Factor the common factor from both groups

We can see that \((x-1)\) is the common factor for both groups, so we can factor it out: $$ (x - 1)(3a - 12) $$ This is the factored form of the given polynomial.

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors, which are polynomials of lower degrees. Think of it as breaking down a complex expression into simpler pieces, much like finding the prime factors of a number. For example, the expression \(x^2 - 1\) can be factored into \(x+1\) and \(x-1\), which are its factors. This method is incredibly useful in solving equations, simplifying expressions, and understanding the properties of graphs. When dealing with polynomials of higher degrees or multiple terms, techniques like the grouping method become essential tools for factorization.

Grouping Method

The grouping method is a strategy for factoring polynomials that involves rearranging the terms into groups that have common factors. Here's a simplified roadmap to navigate through this method:

• Rewrite the polynomial by grouping terms with common factors.
• Factor out any common factors from each group.
• Look for and factor out any common binomial factors between the groups.

If done correctly, the result is a simpler expression that reveals insights into the solutions of an equation or zeroes of a polynomial function. Utilizing the grouping method often involves several steps of rearranging and combining like terms, as demonstrated in the exercise above.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a specific value when the variables are substituted with real numbers. They form the cornerstone of algebra and can range from simple expressions like \(3x+2\) to more complex ones involving exponents, roots, and multiple terms. Understanding how to manipulate these expressions using algebraic operations is crucial in solving equations and inequalities. Mastery of algebraic expressions enables students to transition from arithmetic to algebra, where problems become more about the relationships between numbers rather than computation alone.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with the properties of operations on numbers and the construction and evaluation of algebraic expressions through the use of variables. It's the stepping stone from basic arithmetic to more advanced topics in mathematics. In elementary algebra, you'll learn foundational concepts such as operations with algebraic expressions, factoring, and solving linear equations. These concepts are not just essential for mathematics; they are used in various fields such as science, engineering, economics, and beyond. Embracing the basics of elementary algebra means building the analytical skills needed to tackle more sophisticated problems.