Problem 57 - Elementary Algebra

Question

Factor. \(x^{2}+x y+5 x+5 y\)

Step-by-Step Solution

Answer

The factored form is \((x + y)(x + 5)\).

Step 1 - Group the terms

First, group the terms in pairs to make it easier to factor. The expression is: \[x^{2} + xy + 5x + 5y\] Group the terms to make it: \[(x^2 + xy) + (5x + 5y)\]

Step 2 - Factor common terms in each group

Factor out the common terms from each group. For the first group, \(x^2 + xy\), the common term is \(x\). For the second group, \(5x + 5y\), the common term is \(5\). This gives us: \[x(x + y) + 5(x + y)\]

Step 3 - Factor by grouping

Now factor out the common binomial factor \((x + y)\) from both terms. This results in: \[(x + y)(x + 5)\]

Key Concepts

Grouping Terms

The first step to factor an algebraic expression like \(\text{x}^{2} + \text{xy} + 5\text{x} + 5\text{y}\) is to group the terms.
This method helps to simplify the factoring process by breaking the expression into smaller and more manageable parts.

In the given expression, we can group the terms into pairs:
(\(x^2 + xy\)) + (\(5x + 5y\)).
Grouping is particularly useful when the expression does not easily show a common factor among all the terms at first glance.

Common Factors

After grouping the terms, the next step is to identify and factor out the common terms within each group.
Let's take a closer look at our grouped terms:

• For the first group, \(x^2 + xy\), the common factor is \(x\). When we factor this out, we get \(x(x + y)\).
• Similarly, for the second group, \(5x + 5y\), the common factor is \(5\). Factoring this out, we get \(5(x + y)\).

By factoring out the common factors from each group, the expression now looks like this: \(x(x + y) + 5(x + y)\).
Notice that both of these factored expressions share a common binomial factor \((x + y)\). Identifying the common factors within each group streamlines the next step in the factoring process.

Factoring by Grouping

In the final step, we utilize the common binomial factor to factor the entire expression by grouping.
From our previous step, the expression is now \(x(x + y) + 5(x + y)\).
Both terms contain the same binomial factor \((x + y)\).
To factor by grouping, we factor out the \((x + y)\) from both terms:
\(x(x + y) + 5(x + y) = (x + y)(x + 5)\).

This gives us the fully factored form of the original expression: \((x + y)(x + 5)\).
Factoring by grouping is a powerful method especially when dealing with quadratic and higher-degree polynomial equations.
It simplifies complex expressions and makes solving equations easier.