Problem 21 - Elementary Algebra

Question

For the following problems, use the grouping method to factor the polynomials. Some polynomials may not. be factorable using the grouping method. $$ 4 p q-7 p+3 q^{2}-21 $$

Step-by-Step Solution

Answer

Answer: No, it is not possible to factor the given polynomial using the grouping method, as the terms inside the brackets after initial grouping and factoring are not the same.

Step 1: Group the terms

We will group the terms in pairs as follows: $$ (4pq - 7p) + (3q^2 - 21) $$

Step 2: Factor out the common factor from each group

We will factor out the common factor from each group: $$ p(4q - 7) + 3( q^2 - 7 ) $$

Step 3: Check if the terms inside the brackets are the same

In this case, we see that the terms inside the brackets are different. The first term inside the brackets is (4q - 7), and the second term is (q^2 - 7). Since the terms inside the brackets differ, we cannot apply the grouping method to factor the polynomial further. So, this polynomial cannot be factored using the grouping method. The final answer is: $$ 4pq - 7p + 3q^2 - 21 $$

Key Concepts

Grouping Method

The grouping method is a handy technique for factoring polynomials, especially when dealing with four terms. The main idea is to rearrange terms in such a way that you can factor something out of each pair, aiming to find a common factor. Let's break down how this works.

• First, divide the polynomial into pairs of terms. This means arranging the expression so that you can clearly see groups—usually with two terms in each group.
• Next, attempt to factor out the greatest common factor from each group. This involves identifying a factor that each term in the pair shares.
• Finally, examine the new expressions formed inside the brackets. You want those to be the same so you can factor them out completely.

Understanding the grouping method involves some practice in identifying suitable pairs and factors. Once mastered, it can simplify seemingly complex algebraic expressions. If the resulting expressions inside parentheses don't match, the grouping method may not work.

Common Factor

A common factor is a value or expression that divides two or more terms evenly. When factoring polynomials, identifying the common factor is an essential step.
To find a common factor, consider the coefficients and variables in each term:

• Look at the numbers in each term. Find the greatest number that divides each exactly—this is your numerical common factor.
• Next, look at the variables. For each distinct variable, determine the lowest power present in all terms—this forms the variable part of your common factor.

Use the common factor to "factor out" from the terms, simplifying the expression. For example, if you have terms like 3x and 6x^2, the common factor is 3x. Factoring it out leads to a simplified form of x(6x + 3).
Finding the common factor is the first move in the group factoring process.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the foundation of algebra and are used to express mathematical relationships. A polynomial is a type of algebraic expression that involves sums and multiples of variables and constants.
When working with algebraic expressions:

• Recognize that terms are parts of the expression separated by addition or subtraction signs.
• Understand that each term can include a variable (like x or y), a numerical coefficient (a number multiplying the variable), and possibly an exponent.
• Learn various methods to manipulate and simplify expressions, such as factoring, expanding, and combining like terms.

The art of working with algebraic expressions lies in knowing which transformations can simplify or solve problems. Factoring, which includes methods like the grouping method, is crucial for reducing polynomial expressions to simpler forms, which in turn makes solving equations easier. Familiarity with these concepts paves the way for solving complex mathematical problems efficiently.