Problem 86 - Introductory Algebra for College Students

Question

Factor by grouping. $$x^{2}+a x+b x+a b$$

Step-by-Step Solution

Answer

The factored form of the expression \(x^{2}+a x+b x+a b\) by grouping is \((x + a)(x + b)\).

Step 1: Identify the Groups

Firstly, we identify the groups. The given expression can be expressed as groups in the following manner:\(x^{2}+a x\) and \(+b x+a b\) are the two groups.

Step 2: Factor Out the Common Factors

Now, factor out the common factor from each group. From the first group \(x^{2}+a x\) we can take out \(x\), which leaves us with \(x(x+a)\). From the second group \(b x+a b\), we can factor out \(b\), giving \(b(x + a)\).

Step 3: Factor Out the Common Binomial

Now, factor out the common binomial. The expression can be rewritten as:\(x(x + a) + b(x + a)\). Here, the common binomial is \((x + a)\). Factoring that out gives us:\((x + a)(x + b)\). This is the factored form of the original expression.

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is crucial when learning algebra. An algebraic expression is a mathematical phrase that can contain numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division.
These expressions often represent real-world problems or situations that can be modeled mathematically by using symbols.

• Variables: Symbols, often letters, that represent unknown values. In our example, the variables used are \(x\) and \(a\).
• Terms: Parts of the expression separated by plus or minus signs. In the expression \(x^{2}+ax+bx+ab\), each separate part is a term.
• Coefficients: Numbers that multiply the variable in any given term. For instance, in \(ax\), \(a\) is the coefficient.

Understanding how these components fit together will aid in manipulating and transforming algebraic expressions more efficiently.

Factoring Techniques

Factoring is a method of breaking down an algebraic expression into simpler components that, when multiplied together, give the original expression. There are several techniques for factoring, but here we focus on the method of "factoring by grouping."This technique is particularly useful when dealing with polynomials that have four terms. Let's break down the strategy:

• Identify groups: Divide the expression into two pairs. For example, the expression \(x^{2}+ax+bx+ab\) can be grouped as \(x^{2}+ax\) and \(bx+ab\).
• Factor each group separately: Look for common factors within each group. The first group \(x^{2}+ax\) has a common factor of \(x\) and the second group \(bx+ab\) has a common factor of \(b\).
• Combine the factored groups: Once you factor out the common factors, you can rearrange the expression. The goal is to find a common binomial factor in both terms, such as \((x+a)\), to simplify further.

This results in a factored expression that's easier to analyze and understand. By practicing this technique, students can improve their problem-solving skills and tackle more complex algebraic problems with confidence.

Introductory Algebra

Learning introductory algebra involves grasping the basic principles and operations of algebra. At this stage, students should become comfortable with manipulating and solving simple algebraic expressions and equations. Some key aspects of introductory algebra include:

• Simplification: Reducing expressions to their simplest form, often through factoring or combining like terms.
• Solving equations: Determining the value of unknown variables that make the equation true.
• Understanding properties: Familiarity with properties of mathematical operations, such as commutative and associative properties, helps in rearranging and simplifying equations more effectively.

These foundational skills are essential as they provide students with the necessary tools to progress to more advanced topics. Factoring, as demonstrated in our solved problem, is a significant part of every introductory algebra course because it's a powerful technique for simplifying expressions and solving polynomial equations. Mastery of these basics will pave the way for success in higher-level math.