Problem 79 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

(a + b)^2

Step 1: Identify Groupings

Separate the polynomial into two groups. For the given polynomial, split it into \(a^2 + ab\) and \(ab + b^2\).

Step 2: Factor Out Common Terms in the First Group

From the first group, \(a^2 + ab\), factor out the common term \(a\). This gives us \(a(a + b)\).

Step 3: Factor Out Common Terms in the Second Group

From the second group, \(ab + b^2\), factor out the common term \(b\). This gives us \(b(a + b)\).

Step 4: Combine the Factored Expressions

Now we have two expressions that both include the common factor \(a + b\). Combine the factored expressions as \((a + b)(a + b)\) or \((a + b)^2\).

Key Concepts

Grouping Method

The grouping method is one of the essential techniques for factoring polynomials, especially when the polynomial does not have more than four terms. It involves breaking down a polynomial into smaller, more manageable groups and then factoring each group separately.
Breaking a polynomial into groups may not always be straightforward, but it becomes simpler with practice. For example, in the polynomial \(a^2 + ab + ab + b^2\), the first group is \(a^2 + ab\), and the second group is \(ab + b^2\). By breaking it down this way, it becomes easier to notice common factors within each group.
Once we have our groups, the next step is to find and factor out the common term in each group.

Common Factor

A common factor is a term that can divide each term in a polynomial evenly. In our example, each group (\(a^2 + ab\) and \(ab + b^2\)) has a common factor. For the first group, \(a^2 + ab\), the common factor is \(a\), because both terms in the group can be divided by \(a\). This gives us \(a(a + b)\) when factored out.
In the second group, \(ab + b^2\), the common factor is \(b\). Both terms can be divided by \(b\), which gives us \(b(a + b)\).
Recognizing and factoring out common factors is crucial as it simplifies polynomials, making them easier to manipulate and solve.

Factoring by Grouping

Factoring by grouping combines the techniques of grouping and factoring out common factors to simplify polynomials. After factoring out the common terms in each group, look for a common binomial factor among the groups. In our example, both groups factor to \(a(a + b)\) and \(b(a + b)\), so the common binomial factor is \(a + b\).
We then write our expressions as the product of this common factor and another binomial. This results in \((a + b)(a + b)\), which can also be written as \((a + b)^2\).
Factoring by grouping not only simplifies the polynomial but also reveals its structure, assisting in solving equations or further algebraic manipulations.