Problem 78 - Elementary and Intermediate Algebra

Question

Factor by grouping. $$ b x+b z+5 x+5 z $$

Step-by-Step Solution

Answer

The factored expression is \((b + 5)(x + z)\).

Step 1: Group the Terms

The expression is \( b x + b z + 5 x + 5 z \). Group the terms into two pairs: \((b x + b z)\) and \((5 x + 5 z)\).

Step 2: Factor the Common Factor from Each Group

For the first group \( (b x + b z) \), the common factor is \( b \), so factor it out: \( b(x + z) \). \(\) For the second group \( (5 x + 5 z) \), the common factor is \( 5 \), so factor it out: \( 5(x + z) \).

Step 3: Factor Out the Common Binomial Factor

Now that both terms have a common factor of \((x + z)\), factor it out: \( (b + 5)(x + z) \).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations, such as addition and multiplication. They don't include an equality sign like an equation does but play a crucial role in forming them.
They can range from very simple, like \( 5x \), to complex, consisting of many terms such as the expression \( bx + bz + 5x + 5z \). In such expressions:

• **Terms** are the distinct parts of an expression separated by addition or subtraction signs. For example, in \( bx + bz + 5x + 5z \), the terms are \( bx \), \( bz \), \( 5x \), and \( 5z \).
• **Coefficients** are the numerical parts attached to the variables. In the term \(5x\), 5 is the coefficient.
• **Variables** are the symbols like \(x\) and \(z\) used to represent unknown values.

Understanding algebraic expressions is essential when factoring, as it helps in recognizing patterns and common factors.

The Nature of Polynomials

Polynomials are specific types of algebraic expressions that consist of terms made up of variables raised to whole number powers and their coefficients. They can be classified by their degree, which is determined by the highest power of the variable present in the expression.
Let's look at these key attributes of polynomials and how they relate to our example expression, \( bx + bz + 5x + 5z \):

• **Degree**: For each term, the degree is the sum of the powers of variables in the term. The whole expression's degree is the highest among its terms. Here, each term is of degree 1.
• **Monomials, Binomials, and Trinomials**: These terms describe polynomials with one, two, or three terms respectively. While the expression given could be broken down into a binomial after factoring \((b+5)(x+z)\).
• **Like Terms**: These are terms within a polynomial that can be combined because they have the same variables raised to the same power. In our original expression, grouping \(bx\) with \(bz\) and \(5x\) with \(5z\) is a strategy for simplifying by identifying like terms.

The process of manipulating polynomials involves adding, subtracting, multiplying, and factoring to simplify or solve algebraic problems.

Identifying and Using Common Factors

A common factor in algebra is a number or variable that divides two or more terms without leaving a remainder. Recognizing and factoring common factors is a pivotal skill in simplifying expressions and solving equations. Let's explore how common factors are identified and utilized, as seen in the example expression.
Here's how you can identify and utilize common factors:

• **Identifying Common Factors**: Look for a factor shared by the coefficients of the terms or a variable common to all terms in a group. In the complete expression \( bx + bz + 5x + 5z \), each subgroup (\(bx + bz\)) and (\(5x + 5z\)) has a common factor, which are \(b\) and \(5\), respectively.
• **Factoring Process**: Once identified, extract the common factor. For example, factoring \(b\) from \(bx + bz\) gives \(b(x + z)\), and similar for \(5(x + z)\) from \(5x + 5z\). The expression thereby becomes \((b+5)(x+z)\) after factoring the binomial \((x+z)\), common to both groups.

This method of factoring by grouping effectively breaks down expressions into simpler components, making them easier to work with, and it's widely used to solve quadratic equations and beyond.