Problem 59 - Elementary Algebra

Question

Factor out the GCF. $$ x 2(4 x-7)-5(4 x-7) $$

Step-by-Step Solution

Answer

The factored expression is \((4x-7)(x^2-5)\).

Step 1: Identify Common Factor

To factor out the greatest common factor (GCF), first look at the entire expression and identify any common terms. In the expression \(x^2(4x-7)-5(4x-7)\), notice that \((4x-7)\) is a term common to both parts of the expression.

Step 2: Factor Out the Common Term

Now, factor out the common term \((4x-7)\) from both parts of the expression. This gives us:\[(4x-7)(x^2 - 5)\].

Step 3: Verify the Factorization

To ensure the factorization is correct, distribute \((4x-7)\) back through \((x^2 - 5)\) and check if it matches the original expression:\[(4x-7)(x^2) - (4x-7)(5) = x^2(4x - 7) - 5(4x - 7)\].The expression is correctly factorized.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a key concept in algebra that helps simplify expressions by factoring them more efficiently. It refers to the largest number or expression that divides two or more terms without leaving a remainder. This process involves identifying common factors shared among terms in a polynomial or an algebraic expression.

To find the GCF, follow these general steps:

• List out all the factors for each term within the expression.
• Identify the common factors shared by the terms you are analyzing.
• Choose the largest of these common factors as your GCF.

In the example given, the GCF is the expression \((4x-7)\), which is common to both \(x^2(4x-7)\) and \(-5(4x-7)\). By factoring it out, you simplify the expression to \((4x-7)(x^2 - 5)\).

Identifying the GCF can streamline solving problems, making further algebraic manipulations easier.

Polynomial Expressions

Polynomial expressions are mathematical phrases that consist of variables and coefficients. Typically, they involve operations of addition, subtraction, and multiplication. They may also have "powers" or "exponents" on the variables. Polynomials can range from very simple to quite complex, depending on the number of terms and the degree of the exponent.

For example, in the expression \(x^2(4x-7) - 5(4x-7)\), both \(x^2(4x-7)\) and \(-5(4x-7)\) are polynomial expressions. Each has a coefficient (\(x^2\) and \(-5\) respectively), a variable term \((4x)\), and a constant \(-7\) within parentheses.

Understanding polynomial expressions involves recognizing each part's role:

• Coefficients: Numbers multiplying a variable (e.g., \(x^2\) in \(x^2 \times (4x-7)\)).
• Variables: Symbols representing unknown numbers (e.g., \(x\)).
• Exponents: Powers applied to variables (e.g., \(x^2\) indicates x is squared).
• Constants: Numbers without variables (e.g., \(-7\)).

Recognizing the structure of polynomial expressions is crucial when performing operations such as factoring, expanding, and simplifying.

Algebraic Expressions

Algebraic expressions form the language of algebra, consisting of variables, numbers, and operations. Unlike polynomials, algebraic expressions can also include division and more complex operations.

They allow us to represent relationships and perform calculations abstractly. Using expressions, we can solve equations or simplify problems, making complex calculations much more manageable.

Key components of algebraic expressions include:

• Terms: Parts of the expression separated by '+' or '-', like \(x^2(4x-7)\) and \(-5(4x-7)\).
• Operators: Symbols that denote operations such as '+' (addition) and '-' (subtraction).
• Brackets: Used to group parts of expressions and manage order of operations, as seen in \((4x-7)\).

In the given exercise, we had to identify algebraic expressions like \((4x-7)\), which appeared in two terms. Factoring this expression helped simplify the entire equation, demonstrating the strength of keeping track of algebraic components.