Problem 6 - Elementary Algebra

Question

Use the grouping method to factor the following polynomials. When factoring the polynomial \(8 a^{2} b^{4}-4 b^{4}+14 a^{2}-7\) in Sample Set A, we grouped together terms 1 and 2 and 3 and 4 . Could we have grouped together terms 1 and 3 and 2 and \(4 ?\) Try this. \(8 a^{2} b^{4}-4 b^{4}+14 a^{2}-7=\)

Step-by-Step Solution

Answer

Answer: Yes, the factored polynomial is \((2a^2 - 1)(4b^4 + 7)\).

Step 1: Group terms 1 and 3 and terms 2 and 4

We rearrange the polynomial as \((8a^2b^4 + 14a^2) + (- 4b^4 - 7)\), thus grouping terms 1 and 3 and terms 2 and 4.

Step 2: Factor each group

We now factor out the common term from each group. The common term for the first group (terms 1 and 3) is \(2a^2\), and the common term for the second group (terms 2 and 4) is \(-1\). The factored expression becomes: \(2a^2(4b^4 + 7)-1(4b^4 + 7)\).

Step 3: Factor the resulting terms

Now we have a common term \((4b^4 + 7)\) in both expressions. So, we can factor this term out of the sum: \((2a^2 - 1)(4b^4 + 7)\). So, yes, we could have grouped together terms 1 and 3 and 2 and 4. The final factored polynomial when considering this grouping is: \((2a^2 - 1)(4b^4 + 7)\).

Key Concepts

Grouping Method

Grouping is a popular technique used in factoring polynomials. It involves rearranging the terms of a polynomial into groups that have a common factor. This method simplifies the process of factoring complex polynomials that may not be easily factorable by other means.
The basic steps include:

• Rearranging the terms: This can involve either grouping terms that share similar factors or arranging them in a way that common factors are evident.
• Factoring each group: Identify a common factor within each group and factor it out, just like you do in simpler expressions.
• Further factoring: This might lead to a common factor between the groups, simplifying the polynomial even more.

Ultimately, grouping helps in organizing the polynomial in such a way that makes it easier to identify patterns and common factors.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler factors that, when multiplied, give the original polynomial. This is a key operation in algebra because it simplifies expressions and allows for easier manipulation.
The process of factoring depends on recognizing patterns and using algebraic principles. For example:

• Identifying common factors: Like numbers, polynomials can have common numerical or variable factors across all terms.
• Using special formulas: Formulas such as the difference of squares or perfect square trinomials can simplify parts of a polynomial quickly.
• Applying methods like grouping: Rearrange and group terms to make factoring more manageable, as demonstrated in the original solution.

These techniques allow for easier computation and offer insights into the structure of polynomials.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. They are the building blocks for more complex mathematical equations and involve using symbols to represent numbers.
Understanding algebraic expressions is essential for working effectively with polynomials. For instance:

• Terms and coefficients: Each part of an expression separated by a plus or minus sign is called a term, and the number accompanying the variable is the coefficient.
• Operations: Just like arithmetic with numbers, we perform operations according to rules of algebra, such as combining like terms and using the order of operations.
• Simplifying expressions: By applying basic algebra rules, expressions can be simplified to reveal their fundamental components and relationships.

Mastering algebraic expressions sets the foundation for engaging with and solving polynomial equations efficiently.