Problem 62 - Algebra A Combined Function

Question

Factor each four-term polynomial by grouping. If this is not possible, write "not factorable by grouping." $$ x y+y+2 x+2 $$

Step-by-Step Solution

Answer

The polynomial factors to \((y + 2)(x + 1)\).

Step 1: Identify Pairs

The polynomial given is \( xy + y + 2x + 2 \). Group the terms into two pairs that can be factored more easily: \((xy + y)\) and \((2x + 2)\).

Step 2: Factor Each Pair

Factor out the common factors in each group. For the first group \((xy + y)\), factor out \(y\) to get \(y(x + 1)\). For the second group \((2x + 2)\), factor out \(2\) to get \(2(x + 1)\).

Step 3: Regroup Using Common Binomial

Notice that \((x + 1)\) is a common factor in both terms \(y(x + 1) + 2(x + 1)\). Factor out \((x + 1)\).

Step 4: Final Factored Form

After factoring out the common binomial \((x + 1)\), the expression becomes \((y + 2)(x + 1)\). Hence, the polynomial can be expressed as the product of \( (y + 2)(x + 1)\).

Key Concepts

factoring by grouping

Factoring by grouping is a technique used to factor polynomials that consist of four or more terms. It's a handy method when direct factoring isn't possible because the polynomial doesn't neatly fit into a standard factorization formula.
To factor by grouping, follow these steps:

• **Identify pairs:** Start by grouping the polynomial's terms into pairs. This may involve looking for terms with common factors or combining those with similar characteristics. In our original example, the terms are grouped as \((xy + y)\) and \((2x + 2)\).


• **Factor each pair:** Examine each pair to find common factors and factor them out. For \((xy + y)\), you can factor out \(y\) to get \(y(x + 1)\). With \((2x + 2)\), factor out \(2\) to get \(2(x + 1)\).


• **Look for common binomials:** After factoring each pair, check if there is a common binomial factor in both groups. In this case, both terms include \((x + 1)\). Factor this out to combine the terms into a simpler product.

The result is a factored expression that's much easier to handle, like \((y + 2)(x + 1)\) in this problem. Factoring by grouping simplifies the polynomial and gives you a clearer view of its structure.

polynomial expressions

Polynomial expressions are algebraic expressions made up of terms that involve variables raised to non-negative integer powers and coefficients. A term in a polynomial could be a constant, a variable, or a product of a constant and a variable raised to a power.
Polynomials are essential in algebra because they let us work with variable expressions in various contexts, ranging from simple algebraic manipulations to complex calculus equations.

• **Terms:** The parts separated by addition or subtraction, such as in \(xy, y, 2x,\) and \(2\).


• **Degree:** The degree of a polynomial is the highest power of the variable present. For a multivariable polynomial like our example \(xy + y + 2x + 2\), the degree is the sum of the powers in the highest-degree term \(xy\).


• **Variable:** A symbol representing an unknown number, commonly denoted by letters such as \(x\) or \(y\).

Understanding the structure and parts of polynomial expressions is key to effectively manipulating and factoring them. Recognizing these elements helps determine the most efficient way to simplify or solve polynomial equations.

algebraic manipulation

Algebraic manipulation involves using rules and techniques to simplify equations or expressions, solving for variables, and finding equivalences. This process is foundational in algebra, helping transform complex problems into more easily solvable forms.
When manipulating algebraic expressions:

• **Combine like terms:** Group terms that have the same variables raised to the same powers. This simplifies calculations and makes it easier to see common factors.


• **Use distributive property:** Apply the distributive law \(a(b + c) = ab + ac\) to expand or factor expressions as needed, as seen when factoring out \(y\) and \(2\) in our example.


• **Factor common elements:** Identify and extract common factors from terms to simplify expressions or prepare for solving equations.

Algebraic manipulation is crucial in working with polynomial expressions since it allows us to explore different forms of the expression and find the simplest or most useful form for a particular problem. Building proficiency with these techniques provides the tools needed for advanced math topics and applications.