Problem 77 - 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

The factored form is (a+b)(a-b).

Step 1: Group the Terms

Group the terms in pairs that share common factors. Hence, group a^{2}+a bdifferently from -a b-b^{2}.

Step 2: Factor out the Common Factors in Each Group

From the first group a^{2}+a bfactor out a,and from the second group, -a b-b^{2},factor out -b.You will get a(a+b)-b(a+b).

Step 3: Factor the Common Binomial

Notice that (a+b) appears in both terms, so factor out the common binomial (a+b).Thus, the expression becomes (a+b)(a-b).

Key Concepts

Factoring by Grouping

Factoring by grouping is a helpful method when dealing with algebraic expressions that do not easily factor straight away. You start by dividing the original expression into smaller groups that have common factors.

For the given problem, the expression is: a^{2} + ab - ab - b^{2}

Start by grouping the terms into pairs:

• Group 1: a^{2} + ab
• Group 2: - ab - b^{2}

This makes it easier to see what each group has in common, making the problem more manageable.

Common Factors

Once you have grouped the terms, the next step is to factor out the greatest common factor (GCF) from each group. The GCF is the largest expression that divides all terms in the group.
For Group 1: a^{2} + ab, the common factor is a. So, you factor out a, getting: a(a + b).
For Group 2: -ab - b^{2}, the common factor is -b. Factor out -b, getting: -b(a + b).

This simplifies our entire expression to: a(a + b) - b(a + b).

Binomial Factorization

After factoring out the common factors, notice that there is a common binomial term (a + b) in both groups.
Factor out this common binomial term:a(a + b) - b(a + b) turns into (a + b)(a - b).

This is called binomial factorization because the expression has been factored into a product of two binomials. This is a key step in simplifying algebraic expressions.

Algebraic Expressions

Algebraic expressions involve variables and constants combined using operations like addition, subtraction, multiplication, and division.
Originally, our expression was a^{2} + ab - ab - b^{2}.
Using techniques like factoring by grouping and identifying common factors helps in simplifying these expressions.
Such methods are essential in making complex algebra problems more straightforward, allowing for easier manipulation and understanding. This way, one can solve further mathematical problems or equations that involve these simplified forms.