Problem 80 - Elementary Algebra

Question

Factor by grouping. Do not combine like terms before factoring. $$ x^{2}+x y+x y+y^{2} $$

Step-by-Step Solution

Answer

(x + y)^{2}

Step 1: Group the Terms

Group the terms in pairs to set up for factoring by grouping. Group \(x^{2} + x y\) and \(x y + y^{2}\). The equation now looks like: \[ (x^{2} + xy) + (xy + y^{2}) \]

Step 2: Factor Out Common Factors

Factor out the greatest common factor (GCF) from each group. From \(x^{2} + x y\), factor out \(x\), and from \(x y + y^{2}\), factor out \(y\): \[ x(x + y) + y(x + y) \]

Step 3: Factor the Common Binomial

Notice that \(x + y\) is a common factor. Factor \(x + y\) out of the entire expression: \[ (x + y)(x + y) \]

Step 4: Write the Final Answer

Express the final factored form of the polynomial: \[ (x + y)^{2} \]

Key Concepts

Polynomial Factoring

Polynomial factoring is the process of breaking down a polynomial into its component factors. It is like reverse-engineering, extracting simpler polynomials that multiply to give the original one. Factoring is useful for solving equations, simplifying expressions, and understanding the roots of a polynomial. For example, our problem deals with the polynomial: \[ x^{2}+x y+x y+y^{2} \] By factoring, we simplified it to \[ (x + y)^{2} \]. This process utilizes methods like factoring by grouping, which is particularly effective when polynomials have four terms. The step-by-step approach makes this complex process much easier to understand and apply.

Greatest Common Factor

The greatest common factor (GCF) is a key concept in polynomial factoring. It is the largest factor that divides each term in a polynomial without leaving a remainder. Finding the GCF helps in simplifying polynomials and is essential in the factoring process.

For example, in the problem:\[ x^{2}+x y+x y+y^{2} \] We first grouped the terms: \[ (x^{2} + xy) + (xy + y^{2}) \].

Then, we factored out the GCF from each group:\[ x(x + y) \] from \( x^{2} + x y \) and \[ y(x + y) \] from \( x y + y^{2} \).

Recognizing the GCF in each group simplifies the polynomial and sets up the next steps in factoring.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and the operations that connect them. They are the building blocks of algebra and appear in various forms, including polynomials. Understanding and manipulating algebraic expressions is the foundation for factoring, solving equations, and graphing functions.

In our exercise, the algebraic expression \( x^{2}+x y+x y+y^{2} \) contains terms that can be grouped and factored. Being familiar with how to group and simplify these expressions makes the factoring process smooth and intuitive.

When you factor algebraic expressions, it's vital to follow structured steps like:

• Grouping terms
• Identifying the GCF
• Factoring out common terms
• Using binomial patterns

By mastering these steps, you can tackle a wide range of problems in algebra with greater ease.