Problem 85 - Elementary Algebra

Question

Either factor out the greatest common factor or factor by grouping. $$ a x+2 y+a y+2 x $$

Step-by-Step Solution

Answer

(a + 2)(x + y)

Step 1: Group the terms

Group the expression into pairs to make it easier to factor. So, you will group \(a x\) with \(a y\) and \(2 y\) with \(2 x\). This gives: \[ (a x + a y) + (2 y + 2 x) \]

Step 2: Factor out the common factors in each group

In the first group \(a x + a y\), factor out \(a\). In the second group \(2 y + 2 x\), factor out \(2\). This gives: \[ a (x + y) + 2 (y + x) \]

Step 3: Combine the like terms

Notice that \(x + y\) and \(y + x\) are the same. Since \((x + y) = (y + x)\), you can factor out \((x + y)\). This gives: \[ (a + 2)(x + y) \]

Key Concepts

greatest common factor

The greatest common factor (GCF) is the largest number or algebraic expression that divides each term in a polynomial without leaving a remainder. To find the GCF of algebraic terms, identify the common elements in each term and select the lowest power of common variables and constants.

For example, in the expression \(ax + ay\), both terms share the factor \(a\). Thus, \(a\) is the GCF.

• Step 1: Identify the common factor in each term (e.g., \(a\) in \(ax\) and \(ay\)).
• Step 2: Factor out the GCF from the expression: \(a(x + y)\).

This process simplifies polynomials and is commonly the first step in factoring more complex expressions.

factoring by grouping

Factoring by grouping is a method used when a polynomial has four or more terms. It involves grouping the terms in pairs and factoring out common factors in each group.

Consider the example: \(ax + 2y + ay + 2x\).
Here are the steps:

• Step 1: Group the terms: \((ax + ay) + (2y + 2x)\).
• Step 2: Factor out the common factors in each group: \(a(x + y) + 2(y + x)\).
• Step 3: Notice that \(x + y\) and \(y + x\) are the same. Factor out \((x + y)\): \((a + 2)(x + y)\).

This results in the factored form \((a + 2)(x + y)\), showing how grouping makes complex polynomials more manageable.

algebraic expressions

Algebraic expressions are combinations of variables, constants, and operations (addition, subtraction, multiplication, and division). Understanding them is crucial for solving and factoring polynomials.

Take the expression \(ax + 2y + ay + 2x\). Each term is formed by a coefficient (a constant or variable) multiplied by a variable (e.g., \(ax\) where \(a\) is the coefficient and \(x\) is the variable).

• Terms: The individual parts separated by plus or minus signs (\(ax\), \(2y\), \(ay\), \(2x\)).
• Coefficients: Numbers or variables multiplying the variable (\(a\), \(2\)).
• Variables: Symbols representing unknown values (\(x\), \(y\)).

Recognizing these components helps simplify and manipulate algebraic expressions, which is essential for operations like polynomial factorization.

polynomial factorization

Polynomial factorization is the process of breaking down a polynomial into simpler factors that, when multiplied together, give the original polynomial. This technique is vital in solving polynomial equations and simplifying expressions.

To factor the polynomial \(ax + 2y + ay + 2x\), follow these steps:

• Step 1: Group the terms to find common factors: \((ax + ay) + (2y + 2x)\).
• Step 2: Factor out the common factors in each group: \(a(x + y) + 2(y + x)\).
• Step 3: Recognize that \(x + y\) and \(y + x\) are the same: \((x + y)\).
• Step 4: Factor \((x + y)\) from each group: \((a + 2)(x + y)\).

This results in the factors \((a + 2)\) and \((x + y)\), which are simpler expressions that explain the structure of the original polynomial. Mastering polynomial factorization is key to tackling advanced algebra problems.